Infrared Aperture Synthesis Imaging of Close Binary Stars ...skraus/papers/kraus.MSthesis.pdf ·...

INFRARED APERTURE SYNTHESIS IMAGING

OF CLOSE BINARY STARS

WITH THE IOTA

A Thesis Presented

by

STEFAN KRAUS

Submitted to the Graduate School of the

University of Massachusetts Amherst in partial fulfillment

of the requirements for the degree of

MASTER OF SCIENCE

Revised Edition February 2004

Department of Astronomy

c© Copyright by Stefan Kraus 2003/04

All Rights Reserved

DEDICATION

Gewidmet Dedicated

meiner Mutter to my Mother



WITH THE IOTA

A Thesis Presented

by

STEFAN KRAUS

Approved as to style and content by:

F. Peter Schloerb, Chair

Ronald Snell, Member

Grant Wilson, Member

Ronald Snell, Department HeadDepartment of Astronomy

ACKNOWLEDGMENTS

Above all, I would like to thank my supervisor ”PETE” SCHLOERB who introduced me into the whole

subject from the very basics. With his pleasant nature, he was always available to give encouragement and

advice which is especially noteworthy since he’s involved in a much larger project. Thanks also for reading

this thesis and giving constructive suggestions for improvements!

This work would have been impossible without the excellent condition of the IOTA facility. So my tribute

to the whole IOTA team!

The complete crew of the astronomy department was very hospitable over the year covering my work.

Graduate Director JOHN KWAN was very keen to help me hurdle all bureaucratic obstacles for my gradua-

tion. My special thanks to him and the entire faculty. An important part of the vital life in the department

are the graduate students who are always good for social activities. Last but not least, best thanks to them

for making my stay such as pleasant!

This work was financially supported by NSF grant AST-0138303.

v

ABSTRACT



WITH THE IOTA

SEPTEMBER 2003

STEFAN KRAUS, VORDIPLOM, RUPRECHT-KARLS-UNIVERSITÄT HEIDELBERG

M.S., UNIVERSITY OF MASSACHUSETTS AMHERST

Directed by: Professor F. Peter Schloerb

We present the first aperture synthesis maps from the Infrared Optical Telescope Array (IOTA) atop Mt.

Hopkins, Arizona. During observation runs in November 2002 and March 2003 PETER SCHLOERB and

myself obtained interferograms of the close binary systems Capella (α Aur) and λ Vir in the H band. This

work covers the whole data reduction process from the raw data to the final maps including different algo-

rithms for visibility and closure phase estimation. A special instrumental effect in our data required the use

of advanced methods for visibility estimation which led to the implementation of a method based on the

continuous wavelet transformation and to the development of a new fringe envelope fitting algorithm. The

results confirm the overall stability and performance of the IOTA facility.

We fitted models of binary stars with uniform disks to our data and were able to track the movement of

the Capella giants over the 14 arc covered by the observed five days in November 2002 with a precision

better than one mas. Together with the data from March 2003 we confirm the orbit published by (Hum94)

and derived uniform disk diameters DAa = (8.4± 0.2) mas; DAb = (6.3± 0.7) mas and the intensity ratio

of the components to be (IAb/IAa)H band = (1.44± 0.23). Using additional information from the literature,

we derived the effective temperatures TAa = (5020+100−90 ) K, TAb = (5730+220

−180) K for the G8 III and G1 III

component. For λ Vir we measured the positions of the unresolved components and determined the intensity

ratio to be (IB/IA)H band = (2.1±0.2).

Aperture synthesis maps of Capella and λ Vir were produced using both the conventional hybrid mapping

and difference mapping imaging strategies. To obtain a superior coverage of the uv plane we performed a

vi

coordinate transformation which compensates the orbital motion of the Capella components. The produced

maps confirm the results from the model fits and converge independently from any model with a resolution

of 5 mas and better.

vii

Contents

Acknowledgments v

Abstract vi

List of Figures xi

List of Tables xiii

1 Introduction 1

1.1 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline of my thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Theory of Long Baseline Interferometry 7

2.1 Basics of Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Effect of Spectral Bandwidth Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Two Sources and the Bandwidth Smearing Effect . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Interferometric Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 The VAN-CITTERT-ZERNIKE Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 Closure Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7 Fundamentals of Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7.1 Model of a Uniform and Limb-darkened Disk . . . . . . . . . . . . . . . . . . . . . 16

2.7.2 Model of a Binary Star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.8 Fundamentals of Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.8.1 Rotation-Compensating Coordinate Transformation . . . . . . . . . . . . . . . . . . 19

viii

3 IOTA Design 21

4 Observations 27

4.1 Observation Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Capella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Data Reduction 32

5.1 Basic Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Properties of our Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.3 From the Raw Data to the Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3.1 Extracting the Fringe Visibility using the Power Spectrum . . . . . . . . . . . . . . 37

5.3.2 Extracting the Fringe Visibility by Fringe Envelope Fitting . . . . . . . . . . . . . . 38

5.3.3 Extracting the Fringe Visibility using the Continuous Wavelet Transform . . . . . . 44

5.4 Correcting Imbalances between the Telescopes . . . . . . . . . . . . . . . . . . . . . . . . 49

5.4.1 Reducing the Matrixfiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.4.2 Applying the Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.5 Extracting the Closure Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Model Fitting 65

6.1 Position of Components in Binary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2 Stellar Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.3 Derived Physical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.4 Wavelength-Dependency of the intensity ratio . . . . . . . . . . . . . . . . . . . . . . . . . 73

7 Aperture Synthesis Mapping 75

7.1 Image Reconstruction Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1.2 Conventional Hybrid Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1.3 Difference Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.1.4 Limiting Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8 Remarks on the Performance of the IOTA Facility 89

ix

9 Conclusions 92

Appendices 94

A - Observation Logs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

B - Data Reduction Software Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Bibliography 104

x

List of Figures

2.1 YOUNG’s double slit experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 MICHELSON stellar interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Shape of the Fringe Package for Binary Sources with different separations . . . . . . . . . . 13

2.4 Definition Closure Phase Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 IOTA at FRED WHIPPLE Observatory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 IOTA dome and siderostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 IOTA beam splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 IONIC3 beam combiners and PICNIC camera . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.1 Sample Fringes: Time Space and Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 Power Spectra with weak and strong Resonance . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3 Sample Fringes with extreme distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.4 Sample Fringes: Fringe Visibility fitted in Power Spectrum . . . . . . . . . . . . . . . . . . 38

5.5 Data November 2002: Fringe Visibility fitted in Power Spectrum . . . . . . . . . . . . . . . 39

5.6 Sample Fringes: Fringe Visibility fitted with Fringe Envelope Algorithm . . . . . . . . . . . 41

5.7 KDE of the Amplitude and Width of the Fringe Envelope . . . . . . . . . . . . . . . . . . . 42

5.8 KS-test of the Fringe Amplitude fitted with the Fringe Envelope Algorithm . . . . . . . . . 42

5.9 The Fringe Envelope Fitting Algorithm under different noise conditions . . . . . . . . . . . 43

5.10 BIAS-Corrected Fringe Amplitude for the Fringe Envelope Fitting Algorithm . . . . . . . . 44

5.11 Data November 2002: Fringe Visibilities fitted with the Fringe Envelope Algorithm . . . . . 45

5.12 Sample Fringes: CWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.13 Sample Fringes: Fringe Visibilities fitted with the CWT Algorithm . . . . . . . . . . . . . . 48

xi

5.14 Data November 2002: Fringe Visibilities fitted with the CWT Algorithm . . . . . . . . . . . 50

5.15 Data March 2002: Fringe Visibilities fitted with the CWT Algorithm . . . . . . . . . . . . . 51

5.16 Sample Matrixfile Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.17 Sample Matrixfile Signal: Nullsignal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.18 Matrixfiles: Star Locker lost Star during Matrixfile Acquisition . . . . . . . . . . . . . . . . 56

5.19 Matrixfiles: Anomalous Signal Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.20 Data November 2002/March 2003: Imbalance Corrections . . . . . . . . . . . . . . . . . . 59

5.21 Data November 2002: Applying the Imbalance Correction to Fringe Visibilities . . . . . . . 60

5.22 Closure Phase: Individual Phases and the Closure Phase . . . . . . . . . . . . . . . . . . . 61

5.23 Frequency Closure versus Closure Phase Histogram . . . . . . . . . . . . . . . . . . . . . . 62

5.24 Data November 2002/March 2003: Fitted Closure Phases . . . . . . . . . . . . . . . . . . . 64

6.1 Stellar Diameter Estimation: α Cas and α Lyn . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.2 Fitted and Reference Positions for Capella . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Model Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.4 Fitted and Reference Positions for Capella . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.1 Dirty Beam and Clean Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.2 Scheme: Hybrid Mapping and Difference Mapping . . . . . . . . . . . . . . . . . . . . . . 78

7.3 Hybrid Map: 2002 November 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.4 Hybrid Map: 2002 November 12..16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.5 Difference Map: 2002 November 12..16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.6 Difference Map: 2002 November 12..16 - Model . . . . . . . . . . . . . . . . . . . . . . . 86

7.7 Hybrid Map: 2002 March 21..24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.8 Mapping the Movement of the Capella Giants . . . . . . . . . . . . . . . . . . . . . . . . . 88

xii

List of Tables

3.1 Allocation of the telescopes to the pixels of the PICNIC Camera . . . . . . . . . . . . . . . 24

4.1 Orbital Elements for Capella (Hum94) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Observed Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1 Equations to correct Imbalance between the Telescopes . . . . . . . . . . . . . . . . . . . . 53

6.1 Stellar Diameters for α Cas and α Lyn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Model Fits for Capella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.3 Model Fits for λ Vir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.4 Derived Physical Parameters for Capella . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.1 Maps of Capella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.2 Maps of λ Vir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

9.1 Observation Log November 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

9.2 Continue: Observation Log November 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9.3 Observation Log March 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

9.4 Continue: Observation Log March 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Chapter 1

Introduction

1.1 Historical Remarks

In the history of astronomy, progress has often been driven by revolutions in instrumentation. It is assumed

that the telescope was invented in the Netherlands in the early 17th century. The first patent application for

a telescope with lenses by HANS LIPPERHEY is dated on October 1608. GALILEO GALILEI (1564-1642)

used such a telescope and published important observations, which were interpreted as strong evidence for

the revolutionary Copernican system. JAMES GREGORY (1638-1675) designed a new type of telescope

which used a mirror instead of lenses, and a few years later SIR ISAAC NEWTON (1642-1727) built the first

of those reflector telescopes. Reflecting telescopes may be built with much larger aperture, which results

in a larger light collecting area so that fainter objects may be observed. Moreover the aperture defines the

ultimate angular resolution θ of a single telescope. Observing at a wavelength λ with a telescope of diameter

D, the effect of diffraction limits the resolution to the RAYLEIGH criterion (BelTh)

θ = 1.22λ/D (1.1)

Currently, the largest optical telescopes have segmented mirrors with diameters D of 10.0m (Keck twins,

Mouna Kea, Hawaii) or monolithic mirrors with 8.2m (VLT, Cerro Paranal, Chile).

Usually, the aperture is not the limiting factor for the angular resolution of a ground-based telescope.

Most telescopes are limited by the atmospheric conditions. The terrestrial atmosphere consists of a turbu-

—1—

Chapter 1 Introduction

lent mixture of regions with different temperatures and densities. This structure is primarily induced by the

cycle of day-night and heating from the ground, and since the refractive index of the atmosphere depends

on its temperature, star light passing through the atmosphere suffers random scattering which distorts the

stellar image. The random nature of this seeing effect produces a Gaussian distributed “seeing disk” which

is determined by a characteristic scale r0 over which the incoming wave front suffers no distortion. This

coherence radius r0 shows a dependence r0 ∼ λ6/5 on the wavelength (Pan98). There is also a coherence

time t0 for which the temporal changing variations can be assumed to be “frozen”. (Bus88) gives typical

values of t0 = 10ms and r0 = 10cm for the V band.

Following (Pan98), the seeing ε depends on r0 as ε = 0.98λ/r0, so even under excellent conditions, the

seeing spreads the image of a point source in the optical to more than 1.1” typically. Therefore, great efforts

have been taken to reduce the influence of the atmosphere. In 1970, a first success was reached by ANTOINE

LABEYRIE with the method of Speckle interferometry. Later this method was improved in the KNOX-

THOMPSON-method and in Speckle Masking Interferometry. Those speckle techniques use extremely short

integration times to “freeze” the atmospheric turbulence, thus limiting the sensitivity of this method for

imaging. Later, accompanied to the technological progress, adaptive optics (AO) systems were built to re-

move atmospheric wave front distortion through the use of simple tip-tilt mirrors as well as more complex

systems using deformable mirrors. The use of AO systems allows the full sensitivity of a telescope to be

achived and these systems have now pushed the resolution of ground-based telescopes even to the diffraction

limit as demonstrated by VLT-MACAO (Hub03) most recently. However, even in the long term there are no

concepts for single-dish-telescopes larger than 100m (see OWL design study). Therefore, a conceptual turn

has to be taken to gain further resolution in optical astronomy.

In 1868, ARMAND FIZEAU (1819-1896) suggested an interferometric method to measure stellar diameters

by placing a mask with two holes in front of a telescope’s aperture. In 1872 and 73, first observations by

EDOUARD STÈPHAN with the 80cm reflector at the Observatoire de Marseille obtained an upper limit of

0.158′′ for the diameter of stars (Qui01). ALBERT ABRAHAM MICHELSON (1852-1931) was the first who

succeded using interferometric methods for astronomical observations. Using the 12′′-telescope on Lick Ob-

servatory he measured the diameters of the Galilean satellites in 1891 (Mic1891). In 1919, he made the first

measurement of the angular diameter of a star (Betelgeuse) with the Hooker Telescope on Mount Wilson

Observatory (Rar). MICHELSON installed a 20-foot beam on top of the 100′′ telescope with two moveable,

flat mirrors. The interferometric pattern (the fringe) had to be located by the observer manually. Then the

—2—

1.2 Outline of my thesis

observer estimated the brightness of this pattern (the visibility). This procedure was repeated for different

baselines. Finally the diameters of seven stars could be measured by MICHELSON and his collaborators.

The first interferometric observation of the close binary star Capella was made by JOHN AUGUSTUS AN-

DERSON in December 1919 (And20). Since the technological challenges to build larger optical interferom-

eters were enormous, interferometry became more and more a branch of radioastronomy in the following

decades. Those attempts in radioastronomy lead to the development of the intensity interferometry method

by HANBURY BROWN and TWISS, which they also applied to optical wavelengths in 1974. In 1974, AN-

TOINE LABEYRIE succeeded in combining the light of two individual telescopes, spaced 12m apart.

Although optical interferometers may be built with large dimensions to achieve high resolution they still are

affected by the earth’s atmosphere. The path length through the atmosphere above each of the telescopes

is constantly changing, which means that although the amplitude of the fringe pattern may be measured,

it’s phase is continually changing. This latter property is an essential part of making an image with an

interferometer by the process known as “aperture synthesis”. Thus, two telescope optical interferometers

are insufficient to carry out imaging experiments. Fortunately, developments of the radio VLBI community

showed that a third telescope in the configuration could yield sufficient new information to make an image,

and these techniques known collectively as Hybrid mapping are now being adapted to optical interferometry.

The first image of the surface of a star was made in 1990 when DAVID BUSCHER et al. (Bus90) presented the

first surface map of Betelgeuse. Another important milestone was achieved in 1995, when JOHN BALDWIN

et al. presented the first optical aperture synthesis map using Capella as the target (Bal96). Those obser-

vations were done with the Cambridge Optical Aperture Synthesis Telescope (COAST) at λ = 830nm. The

longest baseline for this measurement was 6.1m. Four years later, JOHN S. YOUNG presented an infrared

(λ = 1.3µm) map of Capella in his dissertation (You99).

Long baseline interferometry continues to be a very active field with many new instruments coming online.

For this thesis we attempt to achieve the new milestone of creating an aperture synthesis image of Capella

with the Infrared Optical Telescope Array (IOTA).


In 1988, the Smithsonian Astrophysical Observatory (SAO), Harvard University, the University of Mas-

sachusetts (UMass), the University of Wyoming and MIT/Lincoln Laboratory came to the agreement to

build a two-telescope Michelson interferometer with a maximum baseline of 38m. The Infrared Optical

—3—

Chapter 1 Introduction

Telescope Array (IOTA) observed first stellar fringes in December 1993. In 2000 a third telescope com-

pleted the current configuration of IOTA on top of Mount Hopkins, Arizona.

IOTA is one of the few facilities with aperture synthesis capabilities in the near infrared. However, as of

this writing, no aperture synthesis image has been created with this instrument. The goal of my thesis is to

develop a data reduction procedure which gives IOTA those imaging capabilities. To do this, it is important

initially to observe target stars with well known properties such that the final results could be validated with

earlier observations. We decided to choose Capella as our primary target. The orbit of this binary is very

well known (Hum94) and as mentioned earlier there is already a map by COAST (You99). On the other

hand, Capella is more resolved by our interferometer than in any previous synthesis aperture observations

making the final image potentially useful once the performance of this instrument is understood.

This thesis is structured as follows: After the introductory remarks of this chapter, we describe the basic

principles of long baseline interferometry in chapter 2 briefly. Chapter 3 gives an overview of the technical

design of the IOTA observatory and connects the theory of the interferometer to the IOTA facility.

We obtained data on the IOTA site in two observation runs in November 2002 and March 2003, each

lasting one week. In chapter 4 we report details of those runs and the observed targets. Chapter 5 contains

the main part of my work, which is a data reduction procedure to extract the two most important interfero-

metric quantities from the IOTA-scans: The contrast of the fringe pattern (the visibility) and the atmosphere

invariant closure phase. For the visibility estimation, three different methods are examined: One algorithm

is based on the fitting of the fringe envelope to the data in the time domain, another on the fitting of the

fringe power in frequency space and the third on the measurement of the power in the continuous wavelet

transform (CWT) of the data.

In chapter 6 and 7 we interpret those measured quantities scientifically, first by fitting binary star models to

the data and then by generating a model independent hybrid map. The thesis closes with some remarks on

the performance of the IOTA facility in chapter 8 and our conclusions (Chapter 9).

Two appendices contain the detailed observation logs and a short manual for the implementation of my data

reduction software followed by the bibliography.

—4—


All computations were carried out with stand-alone applications, written in C and compiled with gcc-

linux. For graphical visualizations during the computation process PGPLOT was implemented. Fast Fourier

transforms were calculated using FFTW. More details about the applications can be found in Appendix B.

The graphics presented in this thesis were generated with SM, XFig and Matlab. The document itself was

prepared using LaTeX.

—5—

Chapter 2

Theory of Long Baseline Interferometry

2.1 Basics of Interferometry

We understand light as an electromagnetic wave propagating according to the laws of classical electrody-

namics. The most fundamental equations in electrodynamics are the MAXWELL equations, from which one

can derive the HELMHOLTZ wave equation (Weg03). One solution for this equation is the monochromatic,

time-independent planar wave Ψ(~x). Adopting the formalism of (Lüh01), the propagation of this planar

wave through the source-free vacuum can be expressed as

ψ(~x, t) = Ψ(~x)e−ı(ckt+φ) (2.1)

with the wave number k and the relation k = 2πν/c = 2π/λ where c denotes the speed of light in the vacuum,

ν is the frequency and λ the wavelength. φ is the phase of the wave.

With optical detectors we measure not ψ but the absolute square of this complex wavefunction, which is

the intensity I of the light:

I(~x, t) = |ψ(~x, t)|2 (2.2)

Therefore, the superposition of two electromagnetic waves ψ1(~x1, t1) and ψ2(~x2, t2) is given by the absolute

value squared of the sum of both wave functions. Since every measurement needs a finite sampling time, we

—7—

Chapter 2 Theory of Long Baseline Interferometry

have to average over a period much longer than the frequency of the wave, which gives

I12(~x1,~x2, t1, t2) = 〈|ψ1(~x1, t1)+ψ2(~x2, t2)|2〉

= 〈(ψ1(~x1, t1)+ψ2(~x2, t2))(ψ∗1(~x1, t1)+ψ∗

2(~x2, t2))〉

= 〈|ψ1(~x1, t1)|2〉+ 〈|ψ2(~x2, t2)|2〉+2ℜ(〈ψ1(~x1, t1)ψ∗2(~x2, t2)〉)

= I1(~x1, t1)+ I2(~x2, t2)+2ℜ(〈ψ1(~x1, t1)ψ∗2(~x2, t2)〉) (2.3)

where ψ∗ is the complex conjugate of ψ and ℜ(ψ) (ℑ(ψ)) denotes the real (imaginary) part of this complex

function. The last term 〈ψ1(~x1, t1)ψ∗2(~x2, t2)〉 =: Γ12(~x1,~x2, t1, t2) is called the mutual intensity (Lüh01) or

mutual coherence function. Using the assumption of a temporary stationary wave front we define Γ12(~x1,~x2, t1, t2)=:

Γ12(t1 − t2). The mutual coherence function is useful as a measure of the coherence of the signal, but it is

more convenient to use the complex degree of coherence

γ12(t) :=Γ12(t)

√

Γ11(0)Γ22(0)(2.4)

This function is normalized on the intensity as demonstrated clearly in (Lüh01).

If the two waves are uncorrelated, γ12 will vanish. But when the two light waves are coherent, interference

can be observed. This may happen in the case of self-coherence by superposing the same wave at different

times. There are two different cases of self-coherence: For temporal self-coherence the same wave will

interfere at the same place x1 = x2, but at different times t1 and t2. The other case is spatial self-coherence,

where t1 = t2.

Spatial self-coherence is realized in YOUNG’s double slit experiment. This famous experiment has a spe-

cial position in physics since it demonstrates the wave nature of light. Here it may be used to introduce the

basic ideas of a MICHELSON Stellar Interferometer.

The basic setup of YOUNG’s experiment is shown in figure 2.1. Let’s consider a plane, quasi-monochromatic

light wave which is incident from the direction of the unit-vector ~r on an opaque screen with two narrow

slits in it. The width of the slits is assumed to be much smaller than the separation of the slits. The position

of the slits is~x1,~x2 such that the separation can be expressed as ~B =~x1 −~x2. Parallel to the opaque screen, a

detector screen is mounted. From the principle of HUYGENS-FRESNEL it follows that behind the two slits,

the electromagnetic wave will propagate as spherical waves.

—8—


^

d2 d1d2 d1

c= c −

rB

I12

cτ

2 1

’cτ

1

Figure 2.1: Setup of YOUNG’s double slit experiment.

—9—


Assuming that the distance between the two screens is much larger than |B|, basic two-dimensional ge-

ometry lead to the relation

~B~r/c =: B⊥/c =: τ′ (2.5)

where B⊥ is the projected baseline which can be also expressed using the zenith angle z as B⊥ = |~B|cos(z).

The wave front will reach the two slits at different times τ′1 and τ′2. After passing the slits, the rays will

reach the detector screen with different path delays and intensities I1 and I2. The position on the screen can

be expressed in term of an additional optical path delay τ so we define the total path delay τ := τ′2 − τ′1 + τ

for convenience.

Using (2.3), the measured intensity on the detector screen is

I12(τ) = I1 + I2 +2ℜ(

Γ12(τ)eıφ) (2.6)

= I1 + I2 +2√

I1 I2|γ12(τ)|cosφ (2.7)

= I1 + I2 +2√

I1 I2ℜ(

eı(ckτ−φ))

(2.8)

= I1 + I2 +2√

I1 I2 cos(ckτ−φ) (2.9)

Obviously, the pattern on the detector screen is a cosine fringe pattern whose frequency is inversely propor-

tional to the wavelength of the incident frequency.

To separate the varying term in equation (2.9), we define the response R as in (BelTh)

R =I12(τ)− I1 − I2

2√

I1 I2

= cos(2πντ−φ) (2.10)

All concepts introduced in this section for the double slit hold for a Michelson Stellar Interferometer as well.

Figure 2.2 shows that the two telescopes take the position of the slits and the light is not projected on a screen

but combined in a beam combiner in this configuration. Technical details will be discussed in chapter 3. In

the next section the response for a non-monochromatic light will be calculated.

—10—


c/2τ

r

I12 12

B

^

beam combiner

I’

Figure 2.2: A MICHELSON stellar interferometer with a beam combiner (blue) and two detectors (brown).

—11—


2.2 Effect of Spectral Bandwidth Filters

Since astronomical objects radiate over a broad range of the electromagnetic spectrum, it is necessary to

generalize the results from the last section for non-monochromatic light. To obtain a well defined frequency

range it is common to use standardized bandwidth-filters which have a rectangular shape in the ideal case.

Let’s define the rectangular filter with

F(ν) =

1 : ν0 − ∆ν2 < ν < ν0 + ∆ν

2

0 : otherwise(2.11)

So we can just average the response over the bandwidth of F as the transmission function

RF :=

R ∞−∞ RFdνR ∞−∞ dν

=1

∆ν

Z ν0+ ∆ν2

ν0− ∆ν2

Rdν

=1

2π∆ντ−φsin(2πντ−φ)

∣

∣

∣

∣

ν0− ∆ν2

ν0+ ∆ν2

= cos(2πν0τ−φ)1

π∆ντ−φsin(π∆ντ−φ)

≡ cos(2πν0τ−φ)sinc(π∆ντ−φ) (2.12)

With this filter, the fringe will show a cos(τ) modulation and a sinc(π∆ντ) envelope. The envelope is

symmetric around π∆ντ = φ. By measuring the amplitude and position of this central white light fringe the

visibility and the phase can be obtained directly in principle. The white light fringe is defined as the fringe

whose phase is independent of wavelength.

2.3 Two Sources and the Bandwidth Smearing Effect

Now we may introduce a second source. The intensities of the rays I1 and I2 which are incident from different

directions r1, r2 may be different as well. With this setup we have two different total path delays τ1 and τ2

which means when we sum the resulting responses RF,1 and RF,2 together

RF,12 = I1 cos(2πν0τ1 −φ1)sinc(π∆ντ1 −φ1)+ I2 cos(2πν0τ2 −φ2)sinc(π∆ντ2 −φ2) (2.13)

—12—

2.3 Two Sources and the Bandwidth Smearing Effect

a) Unresolved Binary b) Completely Resolved Binary(I1 = 2I2) (I1 = I2)

-100 -50 0 50 100-1

-0.5

0

0.5

1

-100 -50 0 50 100-2

-1

0

1

2

OPD

-100 -50 0 50 100-1

-0.5

0

0.5

1

-100 -50 0 50 100-1

-0.5

0

0.5

1

OPD

c) The Bandwidth Smearing Effect d) Two Fringes(I1 = 2I2) (I1 = 2I2)

-100 -50 0 50 100-1

-0.5

0

0.5

1

-100 -50 0 50 100-1

-0.5

0

0.5

OPD

-100 -50 0 50 100-1

-0.5

0

0.5

1

-100 -50 0 50 100

-0.5

0

0.5

1

OPD

Figure 2.3: Simulated Responses for binary sources with different separations s (s is increasing from the upper left tothe lower right). The individual fringe packages (red and blue) are shown in the upper part of each plot.

—13—


we will get the superposition of two white light fringes which are shifted against each other. Neglecting the

shift due to the phase difference φi j := φi − φ j for a moment, we can examine the dependence of RF,12 on

the angular separation of the two sources s := cos−1(~r1~r2). As soon as B⊥ cos(s) becomes comparable to

k−1 (making the assumption that s ¿ 1 such that B⊥,1 = B⊥,2 ≡ B⊥), the two fringe packages will come out

of phase so they don’t add up perfectly any more. For B⊥ ¿ k, the source is unresolved (see figure 2.3 a)),

otherwise it is resolved. So we obtain an equation comparable to (1.1) using the estimation Bk cosθ ≈ 1

θ ∼ λ/B (2.14)

With the phase difference ∆τ = τ2 − τ1 = 2π and I1 = I2, the source will be completely resolved as shown

in figure 2.3 b). When the difference exceeds 2π significantly, an effect called Bandwidth Smearing can be

observed: The envelope of the resulting fringe package may be deformed significantly. By increasing ∆τ

even more, two separate fringe packages can be observed (see figure 2.3 d)).

From equation (2.14) one may deduce the two common ways to improve the resolution θ of a Michelson

Interferometer: The baseline can be increased and/or a shorter wavelength can be chosen.

2.4 Interferometric Observables

Since the response in (2.13) can vary only in a certain range we may specify the borders of this range:

Rmin = min(RF,12(τ)) and Rmax = max(RF,12(τ)). Finally, the fringe contrast or visibility is defined as

V =Rmax −Rmin

Rmax +Rmin(2.15)

The case with the lowest visibility can be reached by I1 = I2 = I such that 0 ≤V ≤ 1. On the other hand is

V = 1 for an unresolved source.

With the definition of (2.15), we neglect the phase φ21 which obtains additional information about the source.

Therefore it is common to define the complex visibility

= Veiφ21 ∈ (2.16)

Using this elegant formalism, the visibility V is just given by the length of the vector in the complex plane

V = | | while the phase is φ21 = tan−1 (ℜ(

)/ℑ(

)). A very useful property of this formalism is it’s

—14—

2.5 The VAN-CITTERT-ZERNIKE Theorem

additivity.

2.5 The VAN-CITTERT-ZERNIKE Theorem

The VAN-CITTERT-ZERNIKE theorem (Sha92) relates the complex visibility to the Fourier transform F ()

of the irradiance distribution and vice versa

(u,v) =

Z ∞

−∞I(x,y)e−ık(ux+vy)dxdy =: F (I(x,y)) (2.17)

The new coordinates are in frequency space and are defining the uv-plane, where u is the projection of B⊥

onto the x axis and v is the projection of B⊥ onto the y axis and (u,v) are measured in meters.

2.6 Closure Relations

Due to corruption of the signal by the atmosphere, the phases cannot be measured directly from the ground.

ROGERS proposed in 1974 (Weiss) to use the simultaneous phase-measurements of three or more telescopes

to obtain the Closure Phase Φ. Φ is invariant under all atmospheric disturbance as the following considera-

tions show.

Using N telescopes for a observation will result in N(N +1)/2 baselines and the same number of measured

visibilities and phases. But the disturbance of the atmosphere on those baselines is not independent. Each

telescope X may be affected by an arbitrary atmospheric piston which can be described as an additional

delay ζX . Considering the case with N = 3, the measured phase ϕXY between telescope X and Y becomes

ϕAB = φAB +ζA −ζB

ϕBC = φBC +ζB −ζC (2.18)

ϕCA = φCA +ζC −ζA

But the closure phase computed from the measured phases is independent of ζX as in

Φ = ϕAB +ϕBC +ϕCA (2.19)

= φAB +φBC +φCA (2.20)

—15—


where the sign convention is that the baseline vectors close a triangle or loop (see figure 2.4). A necessary

assumption is that this measurement is done within a time interval smaller than the actual coherence time of

the atmosphere so that the phase of the fringe is well defined (see chapter 1). For an array of N telescopes,

(N −1)(N −2)/2 closure phases may be defined which contain a fraction (N −2)/N of the complete phase

information (BelTh). For N > 3 an Amplitude Closure Relation can be obtained to place constraints on the

measured visibility amplitudes as well.

ζA

ζB

ζC

φ

φφ

ABBC

CA

A

C

B

Figure 2.4: Definition of the vector directions in the closure phase triangle.

2.7 Fundamentals of Modelling

It is of practical benefit to derive the visibility and phase functions of simple brightness distributions analyt-

ically. Those functions may be used later to fit models to the measured data. It was shown in section 2.1 that

for a point source V = 1 and Φ = 0.

2.7.1 Model of a Uniform and Limb-darkened Disk

Equation (2.17) can be used to calculate the visibility function for a disk with radius Θ = D/2 and a uniform

brightness distribution I. To simplify the calculations let’s assume a circular stellar disk even if this may be an

inadequate assumption for some rapidly rotating stars (deS03). A parameterization in spherical coordinates

—16—

2.7 Fundamentals of Modelling

leads to

UD(Θ) =

Z Θ

−Θ

Z π

0I(r,ϑ)eıkrB⊥ cosϑrdrdϑ

= 2IUD

Z Θ

0

Z π

0eıkrB⊥ cosϑrdrdϑ

= 2πIUD

Z Θ

0J0(rkB⊥)rdr

=2πIUD

B2⊥k2

Z kΘB⊥

0zJ0(z)dz

= IUD2πΘ2 J1(kΘB⊥)

kΘB⊥(2.21)

with the BESSEL functions of first kind and zeroth (J0) and first order (J1). IUD is the intensity integrated

over the whole disk. Taking the real part (BelTh) and doing the normalization lead to

VUD(Θ) = 2J1(kΘB⊥)

kΘB⊥(2.22)

Of course, the simplification that I(r,ϑ) doesn’t depend on the radius r is unrealistic since most stars are

expected to be limb darkened. Thus a correction for the limb darkening is often applied to the fitted stellar

radii to get the limb darkened diameter ΘLD of a star. The exact value of this correction factor ρ = ΘLD/Θ

depends strongly on the used stellar model so that this problem cannot be solved analytically. (Dav00)

presents a useful correction diagram based on models for five different stellar surface temperatures. For the

H-band this correction factor lies between 1.01 < ρ < 1.03.

There are also empirical approaches like the MICHELSON-PEASE limb darkening function (BelTh) which

express the intensity over a spherical star disk with radius Θ as function of the radius r

I(r/Θ)

I(0)=

[

1−( r

Θ

)2]ξ

(2.23)

It is shown (Hes97) that this powerlaw results in the following visibility function

VLD(Θ) = Γ(ξ−1)2ξ J1(kΘB⊥)

(kΘB⊥)ξ = Γ(ξ−1)

(

2kΘB⊥

)ξ−1

VUD(Θ) (2.24)

which can be fitted to visibility curves.

—17—


2.7.2 Model of a Binary Star

A very useful property of Fourier space is it’s additivity. So V and Φ can be easily derived even for a binary

system with resolved uniform disks.

Let’s consider a binary system with two resolved uniform bright disks and angular radii Θ1,Θ2. The

separation of the two components may be s and the position angle β. It is reasonable to choose for the origin

of the coordinate system the “center of light” of the sources with the brighter (primary) component closer to

the origin. So the distances to the origin become

r1 = sI2

I1 + I2(2.25)

r2 = sI1

I1 + I2(2.26)

Now we transform to the Cartesian coordinate system

~x1 = r1

(

cosβsinβ

)

~x2 = r2

(

cos(β+π)

sin(β+π)

)

(2.27)

this gives the visibility

(~x1,~x2) =

1I1 + I2

( (Θ1)I1e−ik~B~x1 +

(Θ2)I2e−ik~B~x2

)

(2.28)

Now it is straight forward to show that

V 2(~x1,~x2) = V 2(Θ1)+V 2(Θ2)+2V 2(Θ1)V2(Θ2)cos(k~B(~x2 −~x1)) (2.29)

For the measured closure phase one can derive analogous

Φ(~x1,~x2) = tan−1

(

∑XY=AB;BC;CA

I1 cos(k~B~x1)+ I2 cos(k~B~x2)

I1 sin(k~B~x1)+ I2 sin(k~B~x2)

)

(2.30)

For unresolved stars the intensity ratio of a binary star can be calculated from the visibility extrema

(see (BelTh))I1

I2=

1+Vmin

1−Vmin(2.31)

—18—

2.8 Fundamentals of Mapping

2.8 Fundamentals of Mapping

In principle, the VAN-CITTERT-ZERNIKE theorem (2.17) can be inverted to obtain the source intensity dis-

tribution on the sky. However, in practice the map that is produced depends on the sampling of the uv-plane

that is made during data collection. We may define the dirty map to be

I(x,y) =Z ∞

−∞S(u,v)

(u,v)eık(ux+vy)dudv =: F −1(S∗

) (2.32)

where S(u,v) is the sampling function which expresses the coverage of the uv-plane for a particular set of

observations. We can use the Kronecker δ-function to define S in terms of the (u,v)-coordinates of the i’th

of n observations

S(u,v) =n

∑i=1

δ(u−ui)δ(v− vi) (2.33)

Beside the observed points of the uv-plane one may use the fact that the Fourier transform of the sky bright-

ness distribution (as a real function) is hermitian (

(−u,−v) = ∗(u,v)) from which follows V (u,v) =

V (−u,−v). Since the dirty map is a convolution of the real source distribution with the instrument response,

we have to make a deconvolution with the Fourier transform of the sampling distribution defined as the dirty

beam

DB = F −1(S) (2.34)

The dirty map is then given by

DM = CM ∗DB = F −1(S∗ ) (2.35)

so a deconvolution will reconstruct the desired clean map CM. Those relations are discussed in more detail

in (Sch93).

2.8.1 Rotation-Compensating Coordinate Transformation

To obtain a better uv-plane coverage on a binary system with known orbit one may rotate and contract/expand

the (u,v) coordinates in such a way that the movement of the binary components is compensated. Therefore

one has to choose an arbitrary reference position. To keep the correction small we used the position of

the components at the medial time tre f of the observed binary orbit arc as reference point (with separation

s(tre f ) and position angle β(tre f )). Then the transformation from (u,v) to the corrected (u′,v′) coordinates is

—19—


performed in spherical coordinates

α(t) = tan−1(u(t)/v(t))(

u′(t)v′(t)

)

= B⊥s(t)

s(tre f )

(

sin(α(t)− [β(t)−β(tre f )])

cos(α(t)− [β(t)−β(tre f )])

)

(2.36)

—20—

Chapter 3

IOTA Design

Figure 3.1: The Infrared Optical Telescope Array at FRED WHIPPLE Observatory.

The Infrared Optical Telescope Array (IOTA) is located at FRED WHIPPLE Observatory at an elevation of

2,564m (Tra01) atop Mount Hopkins, Arizona. In it’s current configuration, there are three telescopes which

can be moved on an L-shaped track with 15m and 35m long arms. The shorter arm is oriented to the south-

east and carries telescope B. Perpendicular to that, telescope A can be moved along the 35m north-east base-

—21—

Chapter 3 IOTA Design

Figure 3.2: Left: The dome of telescope A in closed position. Right: The siderostat which feeds the CASSEGRAIN

optics (the tube to the left).

line. The third telescope C can be mounted on each of the stations along the north-east baseline. The spacings

between those stations are multiples of 197 inches (= 5.0038m) and 277 inches (= 7.0358m) (Col99), the

length of a IOTA-baseline can be varied between 5m≤ BXY ≤ 38m. Normally one refers to the position of

the telescopes in a rough notation where the position of the telescopes is given in meters on each track (e.

g. A = ne35,B = se15,C = ne10 is the configuration shown in figure 3.1). To define the baselines more

precisely it is necessary to fit the baseline vectors ~BXY = xX − xY for (X ,Y ) = (A,C);(B,C);(A,B) to a

large number of measured delay line offsets (Col99).

The telescopes themselves are CASSEGRAIN telescopes with parabolic f/2.5 45cm primary mirrors which

are fed by 45x86cm siderostats under an angle of 30. The 10x focused parallel beam which leaves the

CASSEGRAIN optics passes a flat mirror whose alignment is controlled by a star tracker. Once the loop

of the star tracker is closed, this tip-tilt adaptive optics uses piezo motors to compensate the atmospheric

induced motion of the image with a frequency comparable to 1/t0. Beside that, it is possible to flip a mirror

in the beam to feed a wide-field (20′) CCD TV-camera whose image can be used for star acquisition.

Next the beam enters an evacuated pipe which contains the two long delay lines (each 28m long). For

telescope C the beam passes the feed and corner mirror and enters the long delay 2 (LD2) from which it

is directed by the exit mirror towards the optical table. The beam from telescope A and B passes the same

configuration of mirrors but the observer may choose which of these is delayed by long delay 1 (LD1).

The configuration with telescope A fixed and telescope B on LD1 is refered to as the South Delayed Case

—22—

Figure 3.3: The beam splitter as seen from the tubes where the beam is leaving the vacuum tanks. The three beam-splitters deflecting the infrared part of the beam to the mirrors on the left where the light is coupled into fibers. Throughthe beam-splitter in the middle we see one of the three mirrors which are deflecting the visual light into the star trackers.

otherwise it is the North Delayed Case. The tracking of the long delays is controlled by the software

automatically (Tra00). Behind the long delays are two short delay lines (SD1, SD2) in the optical path.

They are used to scan carefully through the optical path delay and to locate the fringes. Using the notation

from chapter 2, τ is the delay containing external as well as internal delays. τ is then going to be adjusted

with LD1/LD2 and SD1/SD2 in order to locate the white light fringe at τ = 0.

On the optical table, a beam-splitter separates the visual and infrared components of the beam (see fig-

ure 3.3). The visual light passes the beam-splitter and is finally focused on three different quadrants of

the 32x32 pixel-CCD-chip of the star tracker. The infrared component of the beam is coupled into six op-

tical fibers. An advantage of fiber interferometers is the spatial filtering such that phase irregularities in

the wave-front are converted into amplitude fluctuation that can be corrected for (Mon01). At IOTA, the

light is combined by IONIC3 two-beam combiners (Tra02) which provide in analogy to equation (2.9) the

interferometric output (Mon01)

IXY (τ) = IX + IY +2√

IX IY cos(ckτ−φY X )

I′XY (τ) = IX + IY −2√

IX IY cos(ckτ−φY X ) (3.1)

—23—

Chapter 3 IOTA Design

Figure 3.4: The IONIC3 beam combiners (to the right) and the PICNIC camera (to the left).

Table 3.1: Combinations of the three telescopes on the six pixels of PICNIC

Pixel North Delayed Case South Delayed Case

0 A C B C1 A C B C2 B C A C3 B C A C4 A B B A5 A B B A

where a perfectly balanced coupling is assumed. Using the equations (2.12) and (2.10) it can be shown that

this is equivalent to

IXY (τ) = IX + IY +2V RF√

IX IY

I′XY (τ) = IX + IY −2V RF√

IX IY (3.2)

where a perfectly balanced coupling is assumed.

This interferometric output is focused onto the detector of the PICNIC camera for all three baselines. The

actual allocation of the baselines to the individual PICNIC pixels depends on the configuration North/South

delayed case and can be seen in table 3.1.

—24—

Technical features of this low-noise HgCdTe camera are four individually controlled 128x128 pixel quad-

rants and a high (better than 50%) quantum efficiency in the near infrared (0.8−2.5µm). But most essential

is that the camera is able to measure all six signals within a time frame smaller than the coherence time. It

is reasonable for the observer to adjust some of the camera parameters in dependence of the atmospheric

conditions. The most important camera parameters are Nreads (which is the number of times an individual

pixel is read for averaging), Nloops (which is the number of times a readout is repeated for averaging) and

Nscan (which is the number of readouts during one scan). Under bad atmospheric conditions it might be

reasonable to decrease Nreads.

To scan through the fringe package, the optical delay is changed by the piezos on two baselines. Therefore

the optical path delay (OPD) of a scan is determined by the mechanical stroke L′ of a swipe of the piezo

scanners. Using the equation OPD = 2L′/cos(10) by (Mil99) and the value L′ ≈ 25µm for SD2 from the

headerfiles, then the optical path becomes L ≈ 50µm. SD1 samples with twice that value (≈ 100µm) and the

third baseline with the difference 100µm-50µm= 50µm. So SD1 is scanned at twice the frequency than the

other two baselines.

The camera is also equipped with three broadband filters for the J, H and K’ band (Pedre). For very bright

objects there are neutral density filters with different efficiencies (e. g. ND 3% and ND 25%). In its current

configuration, the limiting magnitude for IOTA to measure fringes is about 7m.0 in the H-band (Pedre). The

six intensities measured on PICNIC (North Delayed Case: [IAC, I′AC, IBC, I′BC, IAB, I′AB]; South Delayed Case:

[IBC, I′BC, IAC, I′AC, IBA, I′BA]) are finally saved in datafiles together with headerfiles. The data reduction of

those files is going to be examined in chapter 5.

—25—

Chapter 4

Observations

4.1 Observation Runs

The data used for this thesis was taken by PETER SCHLOERB and myself in two observation runs, each

lasting about one week. In the first run from the 11th to the 16th of November 2002, we concentrated on the

close binary star Capella. Another goal of this first observation run was to observe calibrator stars in a high

frequency to confirm the overall stability of the instrument. During the week, the atmospheric conditions

were improving continuously up to the last night (2002Nov16) when seeing was worsening again.

The beginning of the second run was strongly affected by bad weather conditions such that data could

only be taken between the 21st and 24th of March 2003. During this run we observed a large variety of

different objects. We placed emphasis on the spectroscopic binary λ Vir for which a period of 206.64d was

measured (Hof82). The same reference refutes reports of an 1.93017d period ((Abt61), (Tok97)). This object

was already observed by some other members of the IOTA-community, so we tried to provide additional data

to allow an orbit fit. Table 4.2 lists the objects and the number of observations which have been done.

The complete observation logs, presented in Appendix A show our observing procedure: Before and

directly after each target star we observed calibrator stars to calibrate resolved objects with point sources (in

the ideal case). When data acquisition on one star is completed, it is part of the observation sequence to take

a set of four additional files with two or all of the beams shuttered out. Later in data reduction, those files

can be used to correct for imbalanced beam combination or other asymmetries.

—27—

Chapter 4 Observations

For most observations we used the H band filter which is mounted on one of the filterwheels of the

PICNIC camera. This standard filter provides a rectangular window centered at λ0 = 1.65µm with width

∆λ = 0.30µm (Mil99). For test purposes we used a 1.65µm narrow band filter as well. A summary of all

observed objects is given in table 4.2.

4.2 Capella

The apparent magnitude of 0m.041 (Kri90) makes Capella the sixth brightest star on the whole sky. Assuming

the parallax of (77.29±0.89) mas measured by Hipparcos (Simbad) one can determine the distance of the

spectroscopic binary system to d = (12.94±0.15) pc.

The system is very well studied. Already the first spectroscopic measurements in 1899 by CAMPBELL (Cam1899)

revealed it’s nature as a binary star. Since then it was studied intensively over the whole accessible range of

the electromagnetic spectrum. The components of the system are classified as G8 III (for Capella Aa) and

G1 III (for Capella Ab) giants (Joh02). More detailed spectroscopic measurements revealed the rotational

periods of the components itself (Str01). The hotter Ab component rotates with (8.64±0.09)d period, which

makes it asynchronous to the binary rotation whereas the cooler Aa component rotates synchronously with

a period of 104.022d (Hum94).

The evolutionary state of the two components is very interesting. Both stars have approximately the

same mass (MAa = (2.69±0.06)M¯; MAb = (2.56±0.04)M¯ (Hum94)) and comparable diameters (rAa =

(8.5±0.1) mas; rAb = (6.4±0.3) mas) but significantly different evolutionary states. Capella Aa is a Helium

burning giant whereas the hotter and more active Ab component is located in the Hertzsprung gap (Joh02).

It is counter-intuitive that the hotter G1 III star and not the late-type component is the more active star. Since

there is strong evidence (Cha96) for this hypothesis, one must expect more starspots and chromospherical

activity on Capella Ab.

The brightness-ratio, IAb/IAa, of the two components in the optical seems to be very close, but not exactly

equal, to one. In the past there were contradictory conclusions about which of the components is the brighter

one (Cha96). Within the last decade high-precision measurements came to the conclusion that Ab is the

brighter component in the visual. Ratios for different wavelenghts are:

—28—

4.2 Capella

Table 4.1: Orbital Elements for Capella (Hum94)

Parameter Capella

a [mas] 56.47±0.05e 0.0000±0.0002

i [] 137.18±0.05Ω (2000.0) [] 40.8±0.1

T0 [JD] 2447528.45±0.02P [days] 104.022±0.002

IAb/IAa(140nm)1 ≈ 0.06

IAb/IAa(275nm)1 ≈ 0.25

IAb/IAa(450nm)2 ≈ 0.77 (∆m = (+0m.28±0m.10))

IAb/IAa(467nm)3 ≈ 0.81 (∆m = (+0m.23))

IAb/IAa(547nm)3 ≈ 0.92 (∆m = (+0m.09))

IAb/IAa(550nm)2 ≈ 0.87 (∆m = (+0m.15±0m.05))

IAb/IAa(643nm)4 ≈ 0.93 (∆m = (+0m.08))

IAb/IAa(800nm)2 ≈ 1.05 (∆m = (−0m.05±0m.05))

IAb/IAa(830nm)5 ≈ 1.11

IAb/IAa(1300nm)6 ≈ 1.4

These measurements will be used later within chapter 6 for comparison with our results.

In the 1920’s ANDERSON and MERRILL derived orbital elements for the α Aur system. A very high

precision orbit of Capella using optical interferometry was provided by (Hum94) (see table 4.2). This orbit

was confirmed by COAST measurements from (Bal96) and (You99) and will be checked for consistency

with the measured IOTA data in chapter 6.

Beside the aperture synthesis maps by COAST in the visual and near-infrared there is an image taken with

the Faint Object Camera on the Hubble Space Telescope which just separates the components at ultraviolet

wavelengths (You02).

1from (You02)2from (Hum94)3from BAGNUOLO & SOWELL 1988 (Hum94)4from STRASSMEIER & FEKEL 1990 (Hum94)5from (Bal96)6from (You99)

—29—

Tabl

e4.

2:O

bser

ved

Obj

ects

.2Θ

:=D

give

sst

ella

rdi

amet

er;a

isth

em

ajor

sem

i-ax

isfo

rbi

nary

syst

ems.

Obj

ect

Oth

erD

esig

natio

nsPu

rpos

eE

xpec

ted

App

eara

nce

No.

ofob

serv

atio

nsN

o.of

diff

eren

tnig

hts

Col

or&

Sym

bol

ofO

bjec

tfor

IOTA

Nov

.’0

2M

arch

’03

Nov

.’0

2M

arch

’03

inpl

ots

αA

urC

apel

laTa

rget

Bin

ary

(a=

56.4

7m

as)1

181

61

blac

k(×

)|

HD

3402

92

com

pone

nts

(4kn

own)

δA

urH

D40

035

Cal

ib.

Dis

kD

=(2

.48±

0.02

6)m

as2

113

42

blue

(×)

βA

urH

D40

183

Cal

ib.

Bin

ary

(a=

3.3±

0.1

mas

)310

-4

-re

d(×

)α

Cas

HD

3712

Cal

ib.

Dis

kD

=(6

.25±

0.31

)m

as4

5-

5-

gree

n(×

)α

Cyg

HD

1973

45C

alib

.D

isk

D=

(2.5

3±

0.61

)m

as4

2-

2-

yello

w(×

)α

Lyn

HD

8049

3C

alib

.D

isk

D=

(9.2

4±

1.02

)m

as4

1-

1-

cyan

(×)

κPe

rH

D19

476

Cal

ib.

Dis

kD

=(2

.78±

0.03

)m

as4

1-

1-

mag

enta

(×)

λV

irH

D12

5337

Targ

etB

inar

y-

11-

4bl

ue(•

)|

2C

ompo

nent

s(3

know

n)H

D12

6035

Cal

ib.

(?)

-17

-4

gree

n(?

)H

D15

8352

Targ

etY

SO-

6-

3cy

an(?

)δ

Crt

HD

9843

0Ta

rget

Dis

k(?

)-

5-

4re

d(•

)H

R21

52C

alib

.(?

)-

2-

1re

d(?

)H

R36

21C

alib

.(?

)-

2-

1bl

ue(?

)H

D15

7856

Cal

ib.

(?)

-2

-2

yello

w(?

)H

D15

9170

Cal

ib.

(?)

-2

-2

mag

enta

(?)

75C

ncH

D78

418

Targ

etB

inar

y-

2-

1m

agen

ta(•

)β

CM

iC

alib

.D

isk

D=

(0.7

9±

0.03

)m

as4

-1

-1

gree

n(•

)λ

Hya

HD

8828

4Ta

rget

Bin

ary

(a=

15.1

8m

as)6

-1

-1

cyan

(•)

41O

phH

D15

6266

Targ

etB

inar

y-

1-

1ye

llow

(•)

1(H

um94

)2

(Bor

02)

3(H

um95

)4

(Ric

02)

5(T

ok97

)6

(Pou

03)

Chapter 5

Data Reduction

5.1 Basic Processing

During data acquisition the scans are written in datafiles, each containing Ndata f ile number of scans. Each of

those scans consists of Nscan readouts each with six pixel values from type double. In November we decided

to save 500 scans in each datafile. In the data reduction process all scans in a datafile are reduced separately

first. After that the quantities from the separate scans are averaged to decrease the statistical error for the

visibility and closure phase measurement. At this point we make the assumption that none of those quantities

is changing significantly during the acquisition of one data file, which lasts typically several minutes. As we

will see later, this assumption doesn’t hold for highly resolved objects. So we choose Ndata f ile = 200 for the

observation run in March.

The PICNIC camera provides an integrated signal over each scan and resets only at the beginning of a scan.

A faster decline in the signal corresponds to a higher flux. By differentiating the raw complementary signals

IXY,raw and I′XY,raw from the baseline between telescope X and Y we obtain IXY =dIXY,raw

dτ and I′XY =dI′XY,raw

dτ .

Then the difference of the two signals is calculated to eliminate the atmospheric “common mode” noise.

To obtain the reduced intensity Ired this difference is normalized. Then the arbitrary offset is removed by

subtracting the mean such that a fringe package will oscillate in the range [−1,+1] around zero. Naturally

this relation doesn’t hold for noise. So with a low signal to noise ratio (SNR) the measured signal may

—32—

5.2 Properties of our Data

exceed this range (see e. g. figure 5.1 b)). To formalize those steps we may write

Ired(τ) =IXY (τ)− I′XY (τ)IXY (τ)+ I′XY (τ)

−〈 IXY (τ)− I′XY (τ)IXY (τ)+ I′XY (τ)

〉 (5.1)

Of course, the sampling by the camera is discrete so we may denote the time between two readouts as δτ.


Figure 5.1 shows some typical scans from our observation run which demonstrate different conditions. The

top row shows nice fringes obtained under good seeing whereas the second row demonstrates worse obser-

vational conditions. Already during data acquisition we mentioned a disturbing effect and will refer to it as

Resonance Effect: Beside the fringe peak there is an arbitrary number of additional peaks at lower and higher

frequencies. To verify that those additional peaks (like in figure 5.1c) are not due to noise we provide power

spectra averaged over 500 scans in figure 5.2. Those peaks are symmetrical around the fringe frequency and

are produced by mechanical resonance of the piezo scanners. The amplitude of this effect is varying over

the whole range from undetectable to completely dominating, but the effect is typically significantly weaker

on the baseline which samples with doubled frequency.

Also in time space one can seen an interesting effect: Sometimes there seem to be fluctuations of the

frequency within the fringe package. Quite often this effect observed in time space may be correlated with

the earlier mentioned resonance problem. Another reasonable explanation is that the atmospheric piston

effect changes the light path above the individual telescopes which may result in frequency changes within

the scan as well. For highly resolved objects like Capella the fringe might be complex due to the nature of

the source structure (see chapter 2.3). An extreme case can be seen in figure 5.3.

The resonance effect described above requires special treatment and data reduction procedures that are

beyond the standard repertoire. For example, to estimate the visibility the routine procedure would be to

measure the power of the fringe in the power spectrum of the scan. However it turned out that this approach

did not lead to satisfying results. Therefore other routines needed to be developed and were implemented.

In the following we present the different procedures and the results demonstrated on the representative data

shown in figure 5.1.

—33—

Chapter 5 Data Reduction

a) Good Conditions (Example: α Cas):

-20 -10 0 10 20

-0.5

0

0.5

2002Nov13/24 - Scan #20 (BC)

0 5 10 15

0

0.5

1

1.5

2

2002Nov13/24 - Scan #20 (BC)

b) Bad Signal-Noise (Example: δ Aur):

-30 -20 -10 0 10 20

-1.5

-1

-0.5

0

0.5

2002Nov13/181 - Scan #30 (BC)

0 5 10 15

0

5

10

2002Nov13/181 - Scan #30 (BC)

c) The “Resonance Effect” (Example: α Lyn):

-20 -10 0 10 20

-0.5

0

0.5

2002Nov13/198 - Scan #45 (BC)

0 5 10 15

0

2

4

6

8

10

2002Nov13/198 - Scan #45 (BC)

Figure 5.1: Representative Scans from November 13, 2002. The left column shows the reduced intensities, the rightone the power spectra. —34—


0 5 10 15

0.02

0.04

0.06

0.08

2002Nov14/200 (AC) - 500 Scans

0 5 10 15

0

0.1

0.2

0.3

0.4

2002Nov13/198 (BC) - 500 Scans

a) b)

Figure 5.2: Averaged power spectra with weak and strong resonance. In the right figure the resonance peaks are evenstronger than the fringe peak.

-20 -10 0 10 20 30

-0.5

0

0.5

-20 -10 0 10 20

-0.5

0

0.5

-40 -20 0 20 40

-0.4

-0.2

0

0.2

0.4

0.6

2002Nov13/29 - Scan #15

0 5 10 15

0

2

4

6

8

10

0 5 10 15

0

2

4

6

8

0 5 10 15

0

2

4

6

2002Nov13/29 - Scan #15

Figure 5.3: Case with extreme distortion in two of the three fringes (Source: α Cas).

—35—


5.3 From the Raw Data to the Visibility

In the following sections we will present different methods to estimate the amplitude of a fringe µ (also

called the Fringe Visibility or the Coherence Factor). To correlate those measured quantities to the visibility

as defined in equation (2.15) we have to correct two important effects:

• The telescopes or the beam combiner might be unbalanced, i. e. the light intensity reaching the

camera pixels from two telescopes may not be equal.

• The interferometric efficiency of the systems must be measured on a reference source. This instru-

mental response is given by the transfer function T and may change during the observation.

Therefore the fringe amplitude must be corrected for both effects to obtain the visibility V :

V =UµT

(5.2)

where U is a correction factor for the imbalance between the two telescopes. The calculation of U will be

discussed in section 5.4.

The value T is defined as the fringe visibility when observing a point source with a balanced system. There-

fore this factor can be easily obtained for one particular time by reducing the observed calibrators in the

same way as the target stars. For an unresolved source T ≡ µcalib,unres. whereas for a resolved calibrator

with an a priori known uniform disk diameter ΘUD the correction becomes T = µcalib,res./VUD(ΘUD) (using

equation (2.22)). Ideally, the calibrator source should be from the same spectral type as the target.

In general, the most promising concept to determine the transfer function for an observation at a particular

time t is the interpolation between neighbouring groups of calibrators. First all contiguous measurements of

one calibrator are grouped and the average 〈T 〉 is calculated for each group. To calibrate a fringe visibility

µ(t) one may just interpolate linearly between the neighbouring calibrator groups. Let’s call the averaged

transfer functions of the two groups 〈Tl(tl)〉 and 〈Tr(tr)〉 then

V (t) = Uµ(t)

(

Tl +Tl −Tr

tl − tr(t − tl)

)−1

(5.3)

Sometimes directly neighbouring calibrator groups might be not available. Then one can make the as-

sumption that T is quite stable over the night. Thus the fringe visibility of all calibrators for that night can

be just averaged to obtain 〈T 〉. Fortunately it can be checked empirically how good the assumption is just by

—36—


looking at the scattering of the estimated fringe visibilities of unresolved stars during that night. This pro-

cedure might be also useful to measure resolved calibrators against other less resolved calibrators to obtain

stellar diameter estimations. This happened in our observation run in November 2002 for the calibrator stars

α Lyn and α Cas. To use them for the calibration of target objects we used the fitted stellar radius Θ and

corrected the measured fringe visibility by dividing with VUD(Θ).

5.3.1 Extracting the Fringe Visibility using the Power Spectrum

According to PARSEVAL’s theorem the power of a signal is the same whether computed in time or frequency

space. Since we may choose an arbitrary normalization, we can define the power spectrum as the absolute

square of the fourier transform P (ν) := |F (ν)|2. In practice, the power spectrum is calculated with a fast

Fourier transform (FFT) and the relation P (ν) := ℜ(F (ν))2 + ℑ(F (ν))2. To avoid aliasing, the data is

multiplied by a window function (e. g. the Welch window (Pre92)) before the computation.

A very nice property of power spectra is that in averaging, real signals add constructively in the power

spectrum while it decreases noise. Thus this method is useful, especially for the measurement of the fringe

power in cases of low SNR.

The width of the fringe peak in the power spectrum is given by the bandwidth but it might also be broad-

ened by atmospheric piston. In addition, there is a noise background which has a positive, but quite constant,

slope in most of the cases. To fit the fringe peak the background is first estimated by measuring the lowest

power Pl(νl) in a window on the lower-frequency end of the covered frequency range. The same is done for

Pr, measured at νr in a window on the high-frequency end of the power spectrum.

This averaged background level with constant slope is subtracted such that mainly the signal remains:

Pbgsub(ν) = P (ν)−(

Pl +Pl −Pr

νl −νr(ν−νl)

)

(5.4)

Finally a Gaussian is fitted to the data. The fit was realized with a least square fit based on the LEVENBERG-

MARQUARDT-algorithm presented in (Pre92). The amplitude, width and central frequency of the Gaussian

are free parameters. With reasonable initial guesses this algorithm works well on data with single peaks as

figure 5.4 a) shows. But the appearance of additional strong resonance peaks can confuse this fitting algo-

rithm and lead to incorrect results (as in figure 5.4 b)).

—37—


0 5 10 15

0.02

0.04

0.06

0.08

2002Nov14/200 (AC) - 500 Scans

0 5 10 15

0

0.1

0.2

0.3

0.4

2002Nov13/198 (BC) - 500 Scans

a) b)

Figure 5.4: Averaged power spectra with weak and strong resonance (black) and the fitted Gaussian with background(red).

The fringe visibility estimated with this method for all nights of our observation run in November 2002

is shown in figure 5.5. It is worth noticing that the abscissa shows the file number and not time as in some

other plots. This way all nights (separated by vertical lines) can be shown in just one plot. The thin hor-

izontal lines show the transfer function averaged over all calibrators observed during one night. Since the

calibrators scatter dramatically about those averages, this method seems not to be adequate to estimate the

fringe visibilities for our data due to the disturbing influence of the resonance peaks.

Also the attempt just to integrate the signal above the noise level within a reasonable-sized window around

the fringe peak frequency (in the hope the total power would be conserved even with resonances) did not

yield satisfying results. Therefore below we consider new methods which do not require use of the power

spectrum.

5.3.2 Extracting the Fringe Visibility by Fringe Envelope Fitting

The Fringe Envelope Fitting algorithm avoids all trouble which may rise from the disturbing resonance or

any piston effects by ignoring the internal structure of the fringe packet. To our knowledge, this new method

—38—


0 200 400 600 800 1000

0 200 400 600 800 1000

File Number

Figure 5.5: The data from our observation run November 2002 reduced using the Power Spectrum Method. The verticallines separate the different nights whereas the horizontal lines give the average of the unresolved calibrator stars. Pointswith negative fitted amplitude were rejected during the reduction process. Color Code: see table4.2

—39—


has not been reported in previous work on this subject. Instead of fitting the response function from equation

(2.12), this algorithm just picks the local maxima and minima and fits this envelope to the sinc-envelope

function.

To determine the fringe envelope, the algorithm runs through all points of a scan and identifies those

points which are local extremes. To ensure that it is not only a small-scale fluctuation, we used for a local

maximum at the point Ired(τ) the criteria Ired(τ)≥ 0 and Ired(τ−2δτ)≤ Ired(τ−δτ) < Ired(τ) > Ired(τ+δτ)≥

Ired(τ+2δτ). By replacing Ired(τ) with −Ired(τ), we get the criterion for the negative envelope points.

Applying these criteria to the data provides a set of points for the positive envelope Epos := (τi, Ired(τi)) as

well as a set of points for the negative envelope Eneg := (τ j, Ired(τ j)). Instead of fitting those two sets of

envelope points separately, we may calculate the set union E := Epos ∪Eneg and fit this to the sinc-envelope

function to obtain a more physically meaningful result since the two envelopes are not independent.

Our algorithm fits the function

Ired(τ) = µ · sinc

(

W2π

(τ−C)

)

(5.5)

To get reasonable results, the fit algorithm needs an initial estimate for the three fit parameters µ, W and C.

For the width W of the envelope one may just use a typical value for this specific baseline. The fringe center

C can be localized by the IOTA-Ames Fringe Tracker (Wil02) with sufficient precision. This program deter-

mines whether or not there is a fringe packet within a scan using the power spectrum and some symmetry

criteria. Additionally it provides an estimate of the position of the fringe and the Gaussian noise. In the

final fit all three parameters may be varied. However the overall result seems to improve when a second fit

is performed with W as fixed parameter, since high noise tends to bias the algorithm to fit wider envelopes.

The χ2 provided by the fit algorithm is used as the criterion to decide whether the fit with free or fixed width

is better.

Figure 5.6 shows the same examples as figure 5.1 but with the extracted envelopes and the fitted sinc

functions. As demonstrated in case c) this algorithm is not affected by the resonance effect. The noise-case

b) shows how important it is for this algorithm to choose reasonable rejection criteria to eliminate noisy

scans. Good criteria are χ2 and the aberration of the fitted values from the estimations. So case b) would be

rejected because of the high χ2 and of the unrealistic large fitted W parameter.

To verify the proper behaviour of this algorithm and justify the step of averaging, we did a Kernel Density

Estimation (KDE) of one group of continuous observations of α Cas. As Figure 5.7 a) shows, is µ for the

—40—



-20 -10 0 10 20

-0.5

0

0.5

2002Nov13/24 - Scan #20 (BC)


-30 -20 -10 0 10 20

-1.5

-1

-0.5

0

0.5

2002Nov13/181 - Scan #30 (BC)


-20 -10 0 10 20

-0.5

0

0.5

2002Nov13/198 - Scan #45 (BC)

Figure 5.6: The same example as in figure5.1 with the extracted envelope (blue) and the fitted envelope function (red).The set of negative envelope points Eneg had be inverted to obtain E !

—41—


a) KDE µ b) KDE W

0.2 0.4 0.6 0.8 1

0.002

0.004

0.006

Fringe Visibility

Fringe Visibility Distribution

40 60 80 100

0

0.005

0.01

0.015

0.02

Fringe Width

Fringe Width Distribution

Figure 5.7: Distribution of the fitted µ and W for 6500 scans from 2002Nov13/21..37 of the resolved star α Cas on theBC baseline. A Gaussian Kernel with width 4.0 was used for the µ-KDE. For the W -KDE the Kernel width was 0.1.

0 200 400 600

0

0.2

0.4

0.6

0.8

1

KS test

Figure 5.8: Results of a KS-test with the Null hypothesis that the distribution of the fitted amplitudes is Gaussian (1=itis Gaussian, 0=it is not Gaussian). 870 data sets are sampled. The colors are representing different channels.

—42—


processed 6500 scans quite Gaussian distributed. Also the distribution of W seems to be well behaved as

figure 5.7 b) shows. A Kolmogorov-Smirnov (KS) test (Pre92) gave the result that for 42 of 870 data sets

the Null hypothesis, that the fringe visibility distribution is Gaussian, must be rejected on a 3σ-level (see

figure 5.8). So it seems to be justified to take the mean as an estimation of µ for a whole datafile. The error

is then given by the standard deviation for N independent scans σ/√

(N −1).

a) Linear Scaling b) Logarithmic Scaling

0 0.2 0.4 0.6

0.2

0.4

0.6

0.8

1

Additive Noise Amplitude

0 0.2 0.4 0.60.1

1


Figure 5.9: Response of the fringe envelope fitting algorithm to additive Gaussian noise. Simulated fringes with am-plitudes of 0.2 (green), 0.4 (blue), 0.6 (red) and 0.8 (black) were generated and then fitted in the same way as real data.The ordinate gives the amplitude of the added noise.

One might expect that the algorithm would tend to overestimate the Fringe Visibility as soon as the noise

level becomes high. To examine the behaviour of the algorithm under controlled conditions we simulated

fringes with additive Gaussian noise of amplitude ε. The typical result for such a simulation can be seen

in figure 5.9: The fitted Fringe Amplitude stays quite stable for low noise and shows then an inclining bias

till the envelope cannot be fitted anymore. Since the bias seems to follow a comprehensible law we tried

to apply corrections. Two different attempts were taken: Since the logarithmic plot in figure 5.9 b) shows

straight lines, we tried first a simple exponential law for the correction

µunbias = µ · e−ερ (5.6)

where ε is provided by the weight-parameter from the IOTA-Ames Fringe Tracker (Wil02) and ρ must be

—43—


0 0.2 0.4 0.6

0.2

0.4

0.6

0.8

1


Figure 5.10: Same data as in 5.9 with the correction from equation (5.6) and the parameter η = 0.5 applied. The noiseε was measured independently from the noise value used for the generation of the Gaussian noise.

determined from the average slope of the lines in figure 5.9 b). The result of this correction can be seen

in figure 5.10. The bias is removed up to a few percent. Since the slopes are not the same for all fringe

visibilities some residuals remain. Therefore a correction was attempted using a 2-D correction matrix

which would supply the correction factor for each unique noise versus fitted fringe amplitude combination.

This matrix was calculated with the generation of simulated fringes for thousands of different parameters.

Unfortunately the corrections from this matrix tend not to be very smooth so we observed undesirable steps

within the corrected fringe visibilities and rejected this method.

Figure 5.11 shows the fringe envelope fitting algorithm applied on the data from November. Bias-

corrections using equation (5.6) were done. The scattering of the calibrator stars around the average value

is quite low and it is easy to perceive that the stars α Cas (green) and α Lyn are resolved compared to other

calibrators.

5.3.3 Extracting the Fringe Visibility using the Continuous Wavelet Transform

The wavelet transformation is a common tool in some special branches of medicine, meteorology and com-

puter science where it is used e. g. for arrythmia diagnostics, the analysis of El Niño phenomena and for

video signal compression. We found the first mention of wavelets for the purposes of astronomical interfer-

ometry in (Que02) where DAMIEN SÉGRANSAN mentions this method for the use of the reduction of noisy

interferograms. Since no further details are given and this method appears only as a footnote in the journals,

—44—


0 200 400 600 800 1000

0 200 400 600 800 1000

File Number

Figure 5.11: The data from our observation run November 2002 reduced using the Envelope Fitting Algorithm. Samesettings as in figure5.5. Color Code: see table 4.2

—45—


we developed our own reduction procedure based on the CWT. In the mathematical formulation of wavelet

theory we follow (Tor98). Unfortunately, for the CWT, there is no analogue to PARSEVAL’s theorem for

Fourier analysis.

In wavelet analysis, the time signal is decomposed into the two-dimensional time-frequency space which

allows localization of the signal. The one-dimensional Fourier transform (FT) measures a signal against

periodic, but non-localized sine and cosine waves, therefore it is not possible to localize the power within

the signal. But instead of just shifting a window over the data and then calculating the FT (what is done in

the windowed Fourier transform WFT), the wavelet transform replaces the sinusoidal waves of the FT with

a function called the mother wavelet ψ(η). The choice of this mother wavelet is quite arbitrary even if this

function has to comply with some normalization conditions (see (Tor98)). Since we are going to measure

the response of this mother wavelet to our interferograms, it is definitely advantageous to choose a mother

wavelet which shows similarities to the fringe function (equation 2.12). Therefore a proper choice might be

the Morlet wavelet, which is a sine wave modulated with a Gaussian

ψ(η) = π−1/4eik0η−η2/2 (5.7)

where k0 is the wavenumber to be adjusted such that the number of fluctuations within ψ roughly fits the

number of fluctuations within a typical fringe package (so we choose k0 = 6).

Now the CWT is defined as a convolution of the signal Ired(τ) with the complex conjugate of a wavelet

(which is just a translated and dilated/contracted version of the mother wavelet)

(τ,s) =

1√s

∞Z

−∞

Ired(t)ψ∗(

τ− ts

)

dt (5.8)

By varying s and τ one obtains a two-dimensional image with real and imaginary part. It is worth mentioning

that the scale s should not be identified with the frequency since the time frequency resolution depends on

the scale s! Finally, we obtain the wavelet power spectrum with PW (τ,s) := | (τ,s)|2.

Since the direct computation of the integral (or summation) in equation 5.8 is very time consuming we

implemented the algorithm by (TorCo) which makes use of the FFT to accelerate the computation. Since we

—46—



OPD

Sca

le

−100 −50 0 50 100

0

20

40

60

80

100

120

OPD

Sca

le

−100 −50 0 50 100

0

20

40

60

80

100

120


OPD

Sca

le

−100 −50 0 50 100

0

20

40

60

80

100

120

OPD

Sca

le

−100 −50 0 50 100

0

20

40

60

80

100

120


OPD

Sca

le

−100 −50 0 50 100

0

20

40

60

80

100

120

OPD

Sca

le

−100 −50 0 50 100

0

20

40

60

80

100

120

Figure 5.12: Left: The CWT of the same examples as in figure5.1. Dark red stands for the highest intensity, white andyellow are low intensities. Right: After applying our filter

—47—



-20 -10 0 10 20

-1

-0.5

0

0.5

1

-20 -10 0 10 20

-1

-0.5

0

0.5

1

2002Nov13/24 - Scan #20 (BC)

After CWT Filtering


-30 -20 -10 0 10 20

-1

-0.5

0

0.5

1

-30 -20 -10 0 10 20

-1

-0.5

0

0.5

1

2002Nov13/181 - Scan #30 (BC)

After CWT Filtering


-20 -10 0 10 20

-1

-0.5

0

0.5

1

-20 -10 0 10 20

-1

-0.5

0

0.5

1

2002Nov13/198 - Scan #45 (BC)

After CWT Filtering

Figure 5.13: The examples from figure5.1 after applying our CWT filter algorithm.

—48—

5.4 Correcting Imbalances between the Telescopes

can neglect normalization factors, the inverse transformation is simply given by

Ired(τ) =

∞Z

−∞

ℜ(

(τ,s))ds (5.9)

The CWT of the sample data can be seen in figure 5.12. Resonance as well as noise causes the highly

localized fringe power to spread over a larger area. To isolate the fringe itself we apply a filter which removes

all signal below some significance level (e. g. 60%). In addition we implemented a routine which filters

out all those areas that are separated from the partition with the highest intensity in it. The spread of this

remaining partition is used as a rejection criteria. When the spread along the scale or time axis or the whole

area is too large then the scan is rejected. The same applies when the partition lies too close to the borders.

For good scans, the power of the fringe is estimated as the integral over the partition.

After the filtering it can be also desirable to perform the inverse transformation as given by equation 5.9. This

way one may evaluate how well the algorithm picks the right area around the fringe. The inversed filtered

CWT is shown in figure 5.13, and it seems that our algorithm works excellently in all cases of good as well

as bad SNR. Also in cases with resonance the algorithm picks the fringe much better than the FT algorithm

(compare to figure 5.2). The partition desegmentation routine tends to cut the offshoots of resonance fringes

away. Therefore the routine may tend to underestimate the fringe power in cases with resonances.

In figure 5.14 and 5.15 we present the data from both of our observation runs reduced with the CWT

algorithm. Two groups on November 15, 2002 extend the calibrator levels (BC baseline). Those are the

scans which were obtained with the 1.65µm narrowband filter (see Log in Appendix A) and had to be

rejected due to the lack of calibrators with the same configuration.


Each observation on one target is completed by the acquisition of the four matrixfiles. The purpose of

three of those files is to measure the individual intensities without any interference. This is achieved just

by shuttering out the light of two telescopes alternately. In the following, I will refer to those three files

with M1 (only the shutter for the fixed delay is open), M2 (only the shutter for SD1 is open) and M3

(only the shutter for SD2 is open). The fourth matrixfile (M4) measures (comparable to a darkframe) the

intensity with all shutters closed. Normally only 100 scans are saved in the matrixfile. The PICNIC camera

on IOTA measures six pixels all in all or three pairs with complementary interferometric outputs, each pair

—49—


0 200 400 600 800 1000

0 200 400 600 800 1000

File Number

Figure 5.14: The data from our observation run in November 2002 reduced using the CWT algorithm. Same settings asin figure5.5. Color Code: see table 4.2

—50—


0 200 400 600 800

0 200 400 600 800

File Number

Figure 5.15: The data from our observation run in March 2002 reduced using the CWT algorithm. Same settings as infigure5.5. Color Code: see table 4.2

—51—


corresponding to one baseline combination. As described in chapter 3 the specific allocation of the individual

pixels to the telescopes depends on the configuration (North/South Delayed Case). To keep the following

equations general we will just refer to the notation already used in equation (5.10) such that X and Y denote

two different telescopes which are producing the interferometric output IXY (τ) and I′XY (τ). But in contrast

to the earlier definition we distinguish now between the individual intensities IaX , Ib

X , IaY , Ib

Y as measured by

the pixel pair a, b on the PICNIC camera. Then it yields that

IXY (τ) = IaX + Ia

Y +2V T RF√

IaX Ia

Y

I′XY (τ) = IbX + Ib

Y −2V T RF

√

IbX Ib

Y (5.10)

where V T is the visibility which would be measured by a perfect balanced instrument. Calculating the

reduced intensity as defined in equation 5.1 (but ignoring arbitrary offsets for simplicity) we obtain the

general formulae

Ired =

IaX + Ia

Y − IbX − Ib

Y +2V T RF

(

√

IaX Ia

Y +√

IbX Ib

Y

)

IaX + Ia

Y + IbX + Ib

Y +2V T RF

(

√

IaX Ia

Y −√

IbX Ib

Y

) (5.11)

Normally this relation is simplified with the assumption of a perfect beam combiner ((

IaX = Ib

X ≡ IX)

∧(

IaY = Ib

Y ≡ IY)

and perfectly balanced telescopes IX = IY therefore equation 5.11 could be simplified to

Ired, perf2 = V T RF . Of course, this assumption is not justifiable, instead we should assume a more general

case and separate the V -dependent term in equation (5.11). But it is reasonable to assume at least IaX Ia

Y ≈ IbX Ib

Y ,

so we can make the good approximation

Ired ≈IaX + Ia

Y − IbX − Ib

Y +2V T RF

(

√

IaX Ia

Y +√

IbX Ib

Y

)

IaX + Ia

Y + IbX + Ib

Y

= 2V T RF

√

IaX Ia

Y +√

IbX Ib

Y

IaX + Ia

Y + IbX + Ib

Y

+ constant (5.12)

Since we are using the measured Ired to obtain the fitted fringe visibility µ, we can calibrate our data just by

multiplying µ with a correction factor U to get the corrected visibility V as formalized in equation (5.2) with

—52—


Table 5.1: The correction equations for the different pixel pairs and the different cases.

Reduced Intensity North Delayed Case South Delayed Case

IP0,P1red UAC =

IP0A +IP0

C +IP1A +IP1

C√

IP0A IP0

C +√

IP1A IP1

C

UBC =IP0B +IP0

C +IP1B +IP1

C√

IP0B IP0

C +√

IP1B IP1

C

IP2,P3red UBC =

IP2B +IP2

C +IP3B +IP3

C√

IP2B IP2

C +√

IP3B IP3

C

UAC =IP2A +IP2

C +IP3A +IP3

C√

IP2A IP2

C +√

IP3A IP3

C

IP4,P5red UAB =

IP4A +IP4

B +IP5A +IP5

B√IP4A IP0

B +√

IP1A IP1

B

UBA =IP4B +IP4

A +IP5B +IP5

A√IP4B IP4

A +√

IP5B IP5

A

UXY =IaX + Ia

Y + IbX + Ib

Y√

IaX Ia

Y +√

IbX Ib

Y

(5.13)

In table 5.1, we present the correction equations for all possible cases. The values for all intensities must

be estimated from the matrixfiles, which is the topic of the following subsection.

5.4.1 Reducing the Matrixfiles

The matrixfiles have exactly the same structure as the data files. So the first step in data reduction is again

taking the derivative of the raw data supplied by PICNIC. A more negative signal corresponds to a higher

measured flux on the CCD. Since the light intensity which reaches the camera from the baseline with open

shutter is supposed to be constant for the individual pixels, we expect a stable signal. But as the example

in figure 5.16 shows, there is a small drop in the intensity in the beginning of each scan (enlarged in fig-

ure 5.17). We suppose that this drop is caused by the initial reset of the camera.

Figure 5.16 shows the inequality we want to correct for: The averaged intensity 〈IM1P2 〉 measured on

pixel P2 (blue) in M1 is significantly stronger than the intensity 〈IM3P2 〉 in M3. Possible reasons for these

inequalities will be discussed in chapter 8.

Both of the above mentioned intensities still contain the dark current background due to the instrument

alone. We have to subtract this background to obtain the quantity IP2A as required by the equation in table 5.1

—53—


0 50 100 150 200 250-80

-60

-40

-20

0

Time

M1 - 100 Scans

0 50 100 150 200 250-80

-60

-40

-20

0

Time

M2 - 100 Scans

0 50 100 150 200 250-80

-60

-40

-20

0

Time

M3 - 100 Scans

0 50 100 150 200 250-80

-60

-40

-20

0

Time

M4 - 100 Scans

Figure 5.16: A sample for the signals in the matrixfiles (here 2002Nov14/117..120). Each signal was averaged over100 scans. The abscissa have an arbitrary unit. Different colors represent the different pixels on PICNIC: P0=black,P1=green, P2=blue, P3=cyan, P4=red, P5=magenta

—54—


0 50 100 150 200 250-8

-6

-4

-2

0

2

Time

M1 - 100 Scans

0 50 100 150 200 250-8

-6

-4

-2

0

2

Time

M2 - 100 Scans

0 50 100 150 200 250-8

-6

-4

-2

0

2

Time

M3 - 100 Scans

0 50 100 150 200 250-8

-6

-4

-2

0

2

Time

M4 - 100 Scans

Figure 5.17: Same matrixfiles and options as in figure5.16, but with a different scale on the ordinate.

—55—


0 20 40 60 80 100

0

Scan Number

M1 (2002Nov14/85) - Avg. over 256 Samples

Figure 5.18: The intensity of each of the six pixels (different colors), averaged over all 256 samples within a scan. Theordinate gives the number of the scan within the matrixfile from 2002Nov14/85. It seems that the star light was notfalling on the detector before Scan #65, but the user reacquired the star later during the acquisition of the matrix filemanually. Please notice that the intensity in the “null”-levels (green and black) changes as soon as star light illuminatesthe other four pixels.

0 20 40 60 80 100

0

Scan Number


0 20 40 60 80 100

0

Scan Number


0 20 40 60 80 100

0

Scan Number


Figure 5.19: Same settings as in figure5.18 but with the data from 2002Nov14/256..259. The ratio of flux in the BC(black/green) and BA (red/magenta) baseline between the different matrixfiles is so different that it must be assumedthat the star was not being properly tracked during the acquisition of M2. Even when the algorithm ignores in M2 thescans before #49, the flux ratios between the six signals stays 1 : 9, 1 : 12, 1 : 3.4, 1 : 2.6, 1 : 3.5, 1 : 3.2!

—56—


(for this particular case the equation for the south-delayed case applies). Usually the “null-matrix” M4 would

be used to obtain the average noise level for the individual camera pixels. However we discovered a peculiar

effect which can be seen in figure 5.17: The averaged intensity in the null-matrix is significantly higher

than in the other matrixfiles - even if the baselines are shuttered off in exactly the same way! Extended

investigations showed that this relation always holds (IM4 > IMX for the shuttered out pixels on X = 1,2,3).

There is a very illustrative case which confirms this effect as well: During the acquisition of the matrixfile

shown in figure 5.18 the star was not being tracked by the star-tracker initially and the star light did not reach

the camera pixels during the first 60 scans. Then the star tracker locked onto the star and simultaneously

to the increase of the intensity on two baselines, the background falls off at the completely shuttered out

baseline! Therefore the intensity measured by the six pixels is not independent! Since the intensity drops

it can be ruled out that this behaviour is caused by light scattered from the illuminated pixels onto the dark

pixels. Therefore we suspect the camera electronics as the originator of this undesirable effect. To avoid the

situation that the real signal level may become lower than the background level we do not use the null-matrix

M4 but the background levels in the remaining matrixfiles. So for the South Delayed Case (as in the earlier

presented example) the relations for noise-subtraction become

IP2A = 〈〈IP2

M1〉−〈IP2M2〉〉 IP3

A = 〈〈IP3M1〉−〈IP3

M2〉〉

IP4A = 〈〈IP4

M1〉−〈IP4M3〉〉 IP5

A = 〈〈IP5M1〉−〈IP5

M3〉〉

IP0B = 〈〈IP0

M2〉−〈IP0M1〉〉 IP1

B = 〈〈IP1M2〉−〈IP1

M1〉〉

IP4B = 〈〈IP4

M2〉−〈IP4M3〉〉 IP5

B = 〈〈IP5M2〉−〈IP5

M3〉〉

IP0C = 〈〈IP0

M3〉−〈IP0M1〉〉 IP1

C = 〈〈IP1M3〉−〈IP1

M1〉〉

IP2C = 〈〈IP2

M3〉−〈IP2M2〉〉 IP3

C = 〈〈IP3M3〉−〈IP3

M2〉〉

The all-embracing brackets denote the averaging over the whole 100 scans which are contained in one

matrixfile whereas the inner brackets refer to the averaging over the 256 readouts within one scan. Especially

the averaging over the 100 scans must be done carefully. It may happen that the star is lost by the star

trackers (as in the case in figure 5.18). In those cases, simple averaging will result in an underestimation

of the overall averaged intensity and therefore in a correction factor which is too large! My data reduction

program reconstructs those matrixfiles by separating over those scans which contain clearly signal. When

the star was lost during the whole or a dominant part of the matrixfile acquisition, the matrixfile is rejected

—57—


and the correction factor cannot be calculated. There are other cases like the one from 2002Nov14/256..259

(Figure 5.19) where the flux in the baselines between the different matrixfiles is such as unbalanced (in

this case 1 : 12) that one must suppose that the star was not properly locked by the star tracker during the

acquisition of at least one of the matrixfiles. Therefore we had to choose a threshold ratio beyond which

matrixfiles are rejected (e. g. 1 : 4).

5.4.2 Applying the Correction

When all four averaged intensities are known for one baseline, the correction factor UXY can be calculated

using table 5.1. To demonstrate that for the sample case, we may calculate the specific values for UBC:

IP0B = 18.46 IP1

B = 22.99

IP0C = 19.12 IP1

C = 30.48

so the correction becomes

UBC =22.99+18.46+19.12+30.48√22.99×18.46+

√19.12×30.48

= 1.0175

The other baselines are working analogously. For the North Delayed Case another set of equations compa-

rable to equations 5.14 must be used but those can be easily obtained applying the same underlying concept.

If we recall the fringe visibilities in figures 5.11 and 5.14, obtained with two completely different methods

leading to remarkably similar results, one may come to the conclusion that the remaining small scale (≈ 5%)

fluctuations are the result from imbalances between the telescopes. Therefore we calculated the correction

factors for all baselines and all observations, setting no threshold for extreme flux ratio imbalances (as

discussed in the last section in context with Figure 5.19). The resulting correction factors can be seen in

figure 5.20. Those corrections are too large obviously, so the earlier mentioned problem of unrealistic flux

ratios cannot be neglected (figure 5.19). It seems that in a large fraction of our observations the star tracker

was not able to lock the star properly. But it is also problematic to reject those files and simply to assume

U = 1 because this will just increase the scattering between the measurements. For the following chapters

we used therefore the uncorrected data assuming U = 1 for all scans. Nevertheless we present also the result

when one sets the arbitrary flux ratio threshold for rejection to 1 : 4 (see figure 5.21). Since a large fraction

—58—


0 200 400 600 800 1000

1

1.1

1.2

1.3

1.4Correction factors for obs. run November 2002

File Number

0 200 400

1

1.1

1.2

1.3

1.4Correction factors for obs. run March 2003

File Number

Figure 5.20: Calculated correction factors U for the observation runs in November 2002 (left) and March 2003 (right).The different colors correspond to the three baselines. Some individual values exceeded even the 1.4 border.

of the matrixfiles were rejected, we won’t expect any observable improvement concerning the scattering of

the calibrators compared to figure 5.11.

—59—


0 200 400 600 800 1000

0 200 400 600 800 1000

File Number

Figure 5.21: Fringe Visibilities obtained using the Envelope Fitting method after correction for imbalance betweenpixels. All matrixfiles with flux ratios above 1 : 4 were rejected. For comparison see figure5.11

—60—

5.5 Extracting the Closure Phase


Figure 5.22: The individual phases and the closure phase of the unresolved calibrator δ Aur (left: scan 0..13 and right:scan 72..85) and the resolved binary Capella in between. Data from 2002Nov15/88..169

To estimate the closure phase, we use a quite intuitive method which provides excellent results. As

discussed in chapter 2.6, the closure phase measurement has to be done within a time interval shorter than

the coherence time of the atmosphere t0. Therefore we estimate the positions of the fringes for all three

baselines with the IOTA-Ames Fringe Tracker (Wil02) and average those positions to obtain a position within

the scan which would be most likely to contain high power in all three fringes. Then we set a window around

—61—


0

2500

5000

Figure 5.23: Histogram of Frequency Closure against the measured Closure Phase: The closure phase measurementswhich fulfill the frequency closure relation νclosure = 0 scatter less and show a nice Gaussian distribution. The histogramwas produced using a large number of calibrators: 2002Nov13/166..207 and 2002Nov14/0..54

this averaged position. The size ∆W of this window must be chosen carefully to fulfill the earlier mentioned

condition ∆wδτ < t0 (we choose ∆W = 32 readouts). After calculating the FFT of all three windowed scans,

one can search the position of the peak νXY within the power spectrum (so νXY will be discrete) and calculate

the phases ϕAB, ϕBC, ϕCA with ϕXY = tan−1 [ℑ(F (Ired(νXY )))/ℜ(F (Ired(νXY )))]. The closure phase Φ

is then given by equation 2.20. Figure 5.22 shows the individual phases and the closure phase for one

calibrator-target-calibrator sequence. We expect also a frequency closure relation νclosure = 0 with

νclosure = νAB +νBC +νCA (5.14)

This frequency closure relation can be used to reject individual measurements of lower quality. As fig-

ure 5.23 shows, those points which satisfy the frequency closure relation yield a nice Gaussian distribution.

Therefore our algorithm rejected those measurements where νclosure 6= 0.

Finally the individual closure phase estimations Φi (0 ≤ i ≤ Ndata f ile) for all scans within a datafile are aver-

—62—


aged. To obtain 〈Φ〉 it is important to take the 2π periodicity of the phase into consideration. After bringing

all Φi in the range [0,2π], a proper averaging method would be

〈Φ〉1 =1N

N

∑i=1

Φi

〈Φ〉2 =1N

(

∑∀i:Φi≤π

Φi + ∑∀i:Φi>π

(Φi −2π)

)

σ21 =

1N −1

(

∑(Φ2i )−N(〈Φ〉1)

2)

σ22 =

1N −1

(

∑∀i:Φi≤π

(Φ2i )+ ∑

∀i:Φi>π(Φi −2π)2 −N(〈Φ〉2)

2

)

〈Φ〉 =

〈Φ〉1 : σ1 ≤ σ2

〈Φ〉2 : otherwise(5.15)

The closure phase may show an arbitrary offset so our data reduction software subtracts the average of the

closure phases of all calibrators. The final result for both observation runs can be seen in figure 5.24.

—63—


a) November 2002

0 200 400 600 800 1000

-2

0

2

File Number

b) March 2003

0 200 400 600 800

-2

0

2

File Number

Figure 5.24: Closure Phases from the observation run in November 2002. Measurements which showed an error largerthan 0.2 were rejected. The vertical lines separate the different nights whereas the horizontal lines give the average ofthe unresolved calibrator stars. Color Code: see table 4.2. Due to a change in the configuration (North/South DelayedCase), the closure phase sign changes for the last two nights in November 2002.

—64—

Chapter 6

Model Fitting

6.1 Position of Components in Binary Systems

In order to obtain quantitative information about the observed targets, a first step is to fit simple sky bright-

ness distribution models to the measured visibility and closure phase information. The fundamental math-

ematics for those fits was given in equations (2.22), (2.29) and (2.30). A separate modeling program was

developed which fits binary star models with two uniformly bright disks directly to the output file from the

data reduction software.

Our fit algorithm minimizes χ2 which can depend both on visibilities and closure phases. To weight both

independent quantities equally, we use:

χ2 = 2πχ2V +χ2

Φ

χ2V =

NV

NV +NΦ

NV

∑i=1

(

V 2i −V 2

model

σV

)2

χ2Φ =

NΦ

NV +NΦ

NΦ

∑i=1

(

Φi −Φmodel

σΦ

)2

Including the closure phases into the fit is especially helpful in order to avoid the quadrant error which arises

from the V (u,v) = V (−u,−v) symmetry of the uv-plane. For some nights the closure phase measurement

was poor so we fitted only the visibilities in those cases. Our error estimation for the visibility measurements

include the statistical error as given by the scattering of the individual measurements within each datafile

and an additional calibration error given by the scattering of the calibrators during one night. As a first

—65—

Chapter 6 Model Fitting

Table 6.1: Measured Diameter of resolved calibrators. Data from November 2002 reduced with the Fringe EnvelopeFitting Algorithm.

Object Imbalance Corr. Data Points Fitted Diameter Reference DiameterNV [mas] [mas] (Ric02)

α Cas no 107 5.53±0.12 6.25±0.31| yes 107 6.07±0.11 6.25±0.31

α Lyn no 29 6.82±0.19 9.24±1.02| yes 29 6.96±0.18 9.24±1.02

step, we fixed the stellar radii to fit for the positions of the components, using previously measured radii

from (Hum94) as an initial guess for the Capella giants. For λ Vir we assumed point sources. The fitted

positions are not very sensitive to the intensity ratio (a change in IAb/IAa of ±0.5 results typically in a change

in the position of less than 0.5 mas). This is different for fits with a small number of measurements as for our

one Capella observation in March 2003. In those cases one might not perform a grid search for the global

minima but rather fit with a concrete initial guess.

All nights were fitted separately with the position and intensity ratio as free parameters. Our best-fit re-

sults are listed in table 6.2 and 6.3. The positions may be compared with the predictions by (Hum94) (for

orbital elements see table 4.2). The errors on those fits were estimated by fitting parabolas to the reduced

χ2red of fits with modified intensity ratio (Brute Force Least Square Fit as described in (Schlo)). Calculating

the weighted average and its error obtains the overall best fit intensity ratios of (1.44± 0.23) for Capella

(using Data from November 2002) and (2.06±0.19) for λ Vir.

6.2 Stellar Diameters

Just by fitting the function given in equation (2.22) to the measured visibilities one can estimate stellar

diameters. We did those fits for the Capella giants as well as for the clearly resolved calibrators α Cas and

α Lyn. For the resolved calibrators we had to extrapolate the transfer function since there are typically no

directly neighbouring observations of unresolved calibrators. To obtain the averaged transfer function we

averaged T for all unresolved calibrators from the same nights.

—66—

6.2 Stellar Diameters

a) α Cas b) α Cas with Imbalance Correction

10 15 200

0.2

0.4

0.6

0.8

1

10 15 200

0.2

0.4

0.6

0.8

1

c) α Lyn d) α Lyn with Imbalance Correction

10 15 200

0.2

0.4

0.6

0.8

1

10 15 200

0.2

0.4

0.6

0.8

1

Figure 6.1: Stellar diameter fits for the resolved calibrators α Cas and α Lyn with data taken in November 2002, reducedwith the envelope fitting algorithm. In figure b) and d) the imbalance correction was applied with an arbitrary flux-ratio-threshold of 1 : 4. Different colors indicate measurements on different baselines: blue (AC/BC), red (BC/AC), green(AB/BA) for (North Delayed/South Delayed Case)

—67—

Tabl

e6.

2:Fi

tting

Res

ults

for

Cap

ella

.M

JD=

JD-2

4525

00

Dat

afr

omN

ight

Dat

aPo

ints

χ2 red

Fitte

dI A

b/I

Aa

Fitte

dD

iam

.Fi

tted

Posi

tion1

Ref

.Po

s.1

(Hum

94)

Dat

eM

JDN

VN

Φχ2 V

χ2 ΦD

Aa

DA

bdR

A[m

as]

dDE

C[m

as]

dRA

dDE

C

11/1

2/02

90.8

21...

91.0

4618

052

4.8

4.9

1.28

±0.

628.

4±

1.7

5.6±

2.5

−9.

28±

0.24

44.8

2±

0.11

−8.

9344

.70

11/1

3/02

91.8

05...

91.9

7826

166

5.3

28.1

1.35

±0.

338.

5±

1.1

6.5±

1.4

−12

.81±

0.24

44.1

4±

0.17

−11

.67

43.2

511

/14/

0292

.744

...93

.026

291

105

6.1

32.5

1.28

±0.

588.

3±

0.9

5.8±

1.4

−13

.89±

0.35

41.2

7±

0.18

−14

.47

41.6

011

/15/

0293

.797

...94

.042

369

107

8.3

12.3

1.92

±0.

998.

0±

2.9

8.0±

4.8

−17

.87±

0.30

39.3

7±

0.19

−17

.33

39.7

211

/16/

0294

.883

...95

.034

170

570.

45.

01.

88±

0.62

8.4±

0.8

7.2±

2.0

−20

.12±

0.10

37.7

5±

0.11

−20

.13

37.6

803

/24/

0322

2.67

7...

222.

704

4717

2.04

1.60

±0.

318.

67.

8−

44.3

7±

1.65

−27

.05±

3.12

−46

.40

−28

.23

1R

elat

ive

Posi

tions

are

mea

sure

dfr

omth

ebr

ight

erto

the

fain

ter

com

pone

nt.

Tabl

e6.

3:Fi

tting

Res

ults

for

λV

ir.

MJD

=JD

-245

2500

Dat

afr

omN

ight

Dat

aPo

ints

χ2 red

Fitte

dI B

/IA

Fitte

dPo

sitio

nD

ate

JDN

VN

ΦdR

A[m

as]

dDE

C[m

as]

03/2

1/03

219.

852

...21

9.88

718

859

(2)

−1.

75±

0.60

−18

.07±

0.52

03/2

2/03

220.

855

...22

0.95

011

355

(2)

0.64

±0.

10−

19.3

2±

0.09

03/2

3/03

221.

875

...22

1.95

613

954

4.93

2.18

±0.

44−

0.99

±0.

20−

17.4

0±

0.10

03/2

4/03

222.

828

...22

2.93

720

890

1.01

2.03

±0.

24−

1.09

±0.

01−

17.2

9±

0.01

03/2

1/03

...03

/24/

0321

9.85

2...

222.

937

648

258

3.94

2.04

±0.

08−

1.38

±0.

08−

17.3

9±

0.02

2Si

nce

the

inte

nsity

ratio

sfo

rth

ose

fits

did

notc

onve

rge,

the

fitte

dpo

sitio

nfo

rth

eov

eral

lbes

t-fit

I B/I

A=

2.06

isgi

ven.

6.3 Derived Physical Parameters

Our best fit diameters and statistical errors are given in table 6.1. Figure 6.1 shows the fitted data together

with the visibility curves for our diameter estimations and the measured diameters given by (Ric02). For

α Lyn there is a significant discrepancy between our measurement and the reference measurement. This

may be explained by the lack of a proper imbalance correction as described in chapter 5.4.2. We fitted

also the imbalance corrected data given in figure 5.21 which brought some points in our stellar diameter

fit slightly closer to the reference measurement. It is important to mention that all α Lyn measurements

were taken on the night of 11/13/02 at the break of dawn and are not followed by the measurement of

an unresolved calibrator. Considering those two possible sources of error it is valid to accept a large but

unknown systematical error for the α Lyn measurement. The α Cas measurement agrees with the reference

measurement within the 1σ uncertainty.

For the Capella binary system we fixed the fitted positions and found the diameters with best χ2 again

for visibilities and closure phases simultaneously. The results are listed in table 6.2. After averaging all

nights we obtain diameters of DAa = (8.4±0.2) mas and DAb = (6.3±0.7) mas which is consistent with the

measurements by (Hum94) (DAa = (8.5±0.1) mas and DAb = (6.4±0.3) mas).


Using the results from the last section, one can derive additional physical parameters. With the Hipparcos

measurement of the distance d to Capella (see chapter 4.2) we can calculate the physical radii of the stars to

RAa = (11.7±0.3)R¯ and RAb = (8.8±1.0)R¯.

To obtain an estimate for the effective temperatures we proceed as in (You99) and use the flux ratios

measured by IOTA in the H band and COAST in the J band to obtain the mJ −mH color for each star.

Starting with the fundamental equations

mJAa −mH

Aa = −2.5log

(

IJAa

IHAa

)

mJAb −mH

Ab = −2.5log

(

IJAb

IHAb

)

mJAa+Ab −mH

Aa+Ab = −2.5log

(

IJAa+Ab

IHAa+Ab

)

—69—


0 -20 -40

0

20

40

dRA [mas]

Figure 6.2: Fitted and Reference Positions (Hum94) for Capella: The circles indicate the relative position of the Aacomponent around the fixed Ab component (lower-left corner) along one whole orbit with one day intervals. Thediameter of the circles does not represent the size of the star disks. The red triangles are the position in the middle of theobservated nights and the black triangles with error bars indicate our measurements.

—70—


2002Nov12 2002Nov13 2002Nov14

2002Nov15 2002Nov16

Figure 6.3: Best-fit models for Capella assuming uniform disks with parameters as listed in table 6.2. Statistical andcalibrational errors are shown. The abscissa depicts MJD=JD-2452500 and the CP is given in radians.

—71—


Table 6.4: Derived Physical Parameters for Capella

Parameter Capella Aa Capella Ab

Radius [R¯] 11.7±0.3 8.8±1.0Absolute Brightness V-band [mag] 0m.29±0m.05 0m.18±0m.06

Effective Temperature Teff [K] 5020+100−90 5730+220

−180Bolometric Luminosity L [L¯] 77.0±1.8 74.0±6.5

one obtains

mJAa −mH

Aa = mJAa+Ab −mH

Aa+Ab −2.5log

(

1+ IHAb/IH

Aa

1+ IJAb/IJ

Aa

)

(6.1)

mJAb −mH

Ab = mJAa+Ab −mH

Aa+Ab −2.5log

(

1+ IHAa/IH

Ab

1+ IJAa/IJ

Ab

)

(6.2)

We use the photometric data from (Nog81) with mJAa+Ab =−1m.42 and mH

Aa+Ab =−1m.77 and get mJAa−mH

Aa =

0m.44 and mJAb −mH

Ab = 0m.20. Since there is no error information provided by our resources we cannot pro-

vide error estimations and have to use it as a rough approximation. To relate the infrared colors to effec-

tive temperatures Teff we make use of the work by (Bel89) and get rough estimates for the Capella giants:

Teff,Aa ≈ 5500K and Teff,Ab ≈ 6500K.

The absolute brightness of both components together can be obtained with the relation M = m−5log(d/10pc)

using the apparent brightness in the V band mVAa+Ab = 0.041 (Kri90) which obtains MV

Aa+Ab =−0m.52±0m.03.

Averaging the V band intensity ratio from several observers (see chapter 4.2) gives IVAb/IV

Aa = 0.90±0.03 and

MVAa = 0m.29±0m.05; MV

Ab = 0m.18±0m.06 using the overall luminosity LVAa+Ab = (129.4±3.6)L¯. With the

measured abundance [Fe/H] = 0.43 from (Bri00) and our diameter measurements we can apply the bolo-

metric correction for giant stars from (Alo99) and solve for the effective temperatures and the bolometric

luminosities simultaneously via iteration:

BC =−5.531×10−2

X−0.6177+4.420X −2.669X2

+0.6943X [Fe/H]−0.1071[Fe/H]−8.612×10−3[Fe/H]2 (6.3)

Mbol = MV +BC = −2.5log(L/L¯)+4m.76 (6.4)

L = 4πR2σT 4eff (6.5)

where σ is the STEFAN-BOLTZMANN constant and X = log(Teff)−3.52.

—72—

6.4 Wavelength-Dependency of the intensity ratio

We obtain a bolometric luminosity of the whole system of (151.0± 6.7)L¯ (see table 6.4). Both inde-

pendent methods lead to the result that Ab is the hotter component. Those results fit very well to typical

temperature scales for class III stars. (Alo99) lists typical effective temperatures of (4860±130) K for G8

III stars (as Capella Aa) and (5306±265) K for G2 III, which is close to G1 III (Capella Ab). The large error

for the effective temperature of the Aa component results from our large uncertainty in the stellar diameter.

We did not include errors due to the calibration for the bolometric correction itself as given in equation (6.3).

Our results are in good agreement with (Hum94), who used completely independent resources, other equa-

tions for the bolometric correction and obtained Teff, Aa = (4940± 50) K; Teff, Ab = (5700± 100) K and

LAa = (78.5±1.2)L¯; LAb = (77.6±2.6)L¯.

6.4 Wavelength-Dependency of the intensity ratio

As a result of the intense studies of the system over the last two decades, the intensity ratio of the Capella

giants is known over a broad range of the electromagnetic spectrum. Therefore the spectral dependence

of the intensity ratio can be modelled based on the parameters from our fitting results. As a simple model

one can assume uniform bright disks, each emitting according to Planck’s law like a black body Bλ(T ) of

temperature T. The intensity ratio can then be written as

(

IAa

IAb

)

(λ) =LAaR2

AaBλ(TAa)

LAbR2AbBλ(TAb)

As can be seen in Figure 6.4, this model fits quite well with the earlier reported intensity ratios from the UV

to the K-band as listed in chapter 4.2.

—73—


500 1000 1500 20000

0.5

1

1.5

2

Wavelength [nm]

Figure 6.4: The intensity ratio IAa/IAb of the Capella giants as a function of wavelength. Our own measurement at1650 nm (diamond) is plotted together with other data points from the literature (see text), even if not all references listerrors. The lines show the modelled intensity ratio for two black bodies with the derived best-fit temperatures and stellardiameters as given in table 6.4 (solid line) and the errors taking all uncertainties into account (dashed lines).

—74—

Chapter 7

Aperture Synthesis Mapping

7.1 Image Reconstruction Algorithms

7.1.1 Introduction

The radio astronomy community has developed many strategies to reconstruct images using measured clo-

sure phase information. In this work, we make use of the Conventional Hybrid Mapping (CHM) and the

modified Difference Mapping (DFM) imaging algorithms. The underlying concept of both algorithms is the

iterative self-calibration of the phases using closure phase information and a source model derived from the

CLEAN algorithm applied to a map of the source or of the residuals.

Both algorithms make use of the Clean Beam CB (also Point Spread Function (PSF)) which represents

the response of a point source to the uv-plane sampling obtained in the experiment. In practice we obtained

the CB by fitting ellipses to different contour levels of the dirty beam DB (defined in equation (2.34)). Our

fitting parameters for those ellipses are major and minor semi axes and the rotation angle. Two examples

can be seen in figure 7.1.

7.1.2 Conventional Hybrid Mapping

In CHM we start without any concrete model and set two of the three phases to zero (which is equivalent

to assuming a point source as source model). For each observation, the third phase is determined by the

—75—

Chapter 7 Aperture Synthesis Mapping

2002 November 12..16

-10 -5 0 5 10-10

-5

0

5

10

dRA [mas]

-10 -5 0 5 10-10

-5

0

5

10

dRA [mas]

2003 March 21..24

-10 -5 0 5 10-10

-5

0

5

10

dRA [mas]

-10 -5 0 5 10-10

-5

0

5

10

dRA [mas]

Figure 7.1: Dirty and Clean Beam for two examples. Only the central sector is shown.

—76—


closure phase relation given in equation (2.20). Using the measured visibilities we calculate the DM as

defined by (2.32) and normalize it. In the CLEAN cycle, the weighted DB is subtracted iteratively from

the DM. The weight-factor f for this subtraction is given by a constant gain-factor g times the peak inten-

sity DMmax(xmax,ymax) of the strongest feature within the map f = g ·DMmax(xmax,ymax). The coordinates

(xmax,ymax) and intensities DMmax(xmax,ymax) of those Clean Components CC are stored within a list. After

a constant number Nclean of CLEAN steps are performed, the remaining features represent the residual noise

map NM. By calculating the discrete Fourier transform (DFT) of the CC, the new phases are calculated us-

ing again the closure phase information (Self-Calibration). This procedure is repeated until convergence is

reached. Finally the map may be shifted to the center of light since phase drifting during the hybrid mapping

procedure may result in a movement of the features within the map.

7.1.3 Difference Mapping

The DFM approach tries to support the self-calibration process with an initial guess. This iterative process

adds only positive or negative point sources to the residual map RM. Since not all phases are varied all at

once, this algorithm tends to respond more smoothly and predictably. The difference to CHM is illustrated in

figure 7.2: As an initial model we generate a model map MM which may contain two limb-darkened stellar

disks. Then a DFT is performed in order to simulate the visibilities of this model map. By subtracting those

modeled visibilities from the measured visibilities and performing the DFT we obtain the RM. Applying a

small number of CLEAN steps on the RM will result again in clean components which can be added as δ

functions in the form RMmax(xmax,ymax) ·δ(x−xmax,y−ymax) to the MM. After cleaning, the remaining RM

will be kept as noise map NM. With the modified model the algorithm is repeated until all residuals between

the modeled and measured visibilities have vanished. Finally the MM is convolved with the CB and added

to the NM.

7.1.4 Limiting Factors

Clearly, the advantage of the CHM algorithm is that it is completely model independent. But the proper

convergence behaviour may be affected by the following effects:

a) The brightness distribution is too complex compared to the coverage of the uv-plane.

b) The source is very extended compared to the uv-plane coverage.

—77—


−

New ModelComponents

+

− Vis

*

*

−

+

HYBRID MAPPING

NewPhases

CP

repe

at u

ntil

conv

erge

nce

DIFFERENCE MAPPING

CB

DBDFT

SamplingD

FT

CM

CP

MM

CL

EA

NRM

CM

NM

DFT

DFT

DFT

Model

NM

CL

EA

N

DM

CC

(Vis, CP)Data

Model

repe

at u

ntil

conv

erge

nce

Figure 7.2: Hybrid Mapping and Difference Mapping: CP=(Measured) Closure Phase; Vis=(Measured) Visibilities;DM=Dirty Map; CM=Clean Map; CC=Clean Components; DB=Dirty Beam; CB=Clean Beam; MM=Model Map;RM=Residual Map; DFT=Discrete Fourier Transform

—78—


c) Bandwidth Smearing (see chapter 2.2) may influence the fringe visibility estimations and disturb

proper mapping (may be related to the prior item).

d) Composite spectrum binaries are challenging because the brightness distribution depends not only on

position but also on wavelength.

The items b) to d) may apply for Capella. For λ Vir only item d) may apply. Therefore we expect a better

convergence behaviour for λ Vir which is what we observed in our mapping approaches. To accelerate the

convergence of the Capella maps, one may also set clean windows to limit cleaning of the map to the posi-

tions of the most significant features. Since we are starting with null phases, which is equivalent to a point

source model, a typical behaviour is that most of the power is concentrating in the origin of the uv-plane

first. After a few dozen iterations this power dispenses into strong features at the expected positions of the

stars. Usually less significant features at other places will remain within the map. Setting clean windows

around the main features will assist the convergence of the hybrid map.

Items a) and b) are solved best by increasing the coverage of the uv-plane. To estimate the influence of

Bandwidth Smearing one may compute the CWT for a large number of scans, locate and recenter the center

of the fringe in the individual scans and average all CWTs. This way noise will decrease and real signals

add constructively as in the power spectrum. We simulated typical properties of CWTs which are affected

by Bandwidth Smearing and found the most significant feature to be that in those CWTs the fringe power is

smeared over a broader range in time. Using a reliable source model, the response of the CWT algorithm to

this effect might be simulated and corrected. Since our averaged power spectra apparently did not show this

characteristic fringe peak broadening we did not perform this correction.

One may also correct for d), the Composite Spectrum imaging problem. Simulations by (Hum01) showed

that the quality of maps with stars of significantly different spectral types can be improved noticeably by

introducing two different effective temperatures as additional fit parameters. Naturally, those temperatures

must be assigned to specific regions within the map.

—79—


7.2 Results

Using the introduced mapping procedures we produced a large number of maps. Since the λ Vir data can be

modeled as point sources without any significant residuals, only hybrid mapping seems to be a reasonable

approach. For three of the four nights the uv-plane coverage is very poor which results in a huge, elliptical

CB. Since the modeling in chapter 6.3 confirmed that the stars do not seem to move significantly over the

four days of observation, we added all nights together and obtained a map which easily converges in CHM

without any model assumptions (see figure 7.7).

Since Capella is clearly resolved by IOTA, we produced hybrid maps as well as difference maps of this

object. The data obtained on individual nights with shorter baselines converges without any problems in

CHM due to the simpler brightness distribution. To improve the coverage of the uv-plane, we rotated the

uv-coordinates as described in section 2.8.1. The rotation of the binary components was carried out such

that the components appear at the medial position (dRA(tre f ),dDEC(tre f )) = (14.05,−41.86) mas. To con-

firm that the correction was successful, we repeated the model fit that was used to obtain table 6.2 and got

the overall best-fit position (dRA,dDEC) = (14.35,−40.94) mas with an appreciable improvement in the

overall fit. Using this correction we obtained the hybrid map shown in figure 7.4 and the difference map

in 7.5. Figure 7.3 shows a hybrid map using data from an individual night. For the interpretation of maps

generated with this procedure we should keep in mind that the Aa-component rotates synchronously with

the orbit whereas the brighter and smaller Ab-component rotates with a period of ≈ 8.64d (see chapter 4.2).

Details for all maps presented are given in tables 7.1 and 7.2. We measured the center of light position for

each star to compare the positions of the stars within the maps with the fitted and the predicted positions (as

given in table 6.2). All positions agree very well within expected errors due to the finite beam size. The given

beam sizes equal twice the minor and major axes of the clean beam ellipse measured to the 50% contour.

Also the intensity ratios, obtained by integrating over the area of the individual stars, are in agreement with

the fitted intensity ratios. To estimate the sizes of the resolved stars in the maps we convolved uniform disk

models with the clean beam and measured the response of those models to the stars within the map. To

determine the error on these diameter estimations we performed a Brute Force Least Square Fit (Schlo).

Especially for CHM the diameters are in good agreement with our modeling results as table 7.1 shows.

Such a comparison is important since the produced CHM does not depend on any initial model and therefore

allows us to confirm our earlier results completely independently.

—80—

7.2 Results

We confirmed the movement of the Capella giants along the ≈ 14 arc covered by the five days of our

November observation run in chapter 6.2. Attempts to map the movement of the stars using CHM on data

of individual nights were not successful since the uv-plane coverage is too poor for most of the nights.

Therefore we applied DFM using the same initial model for each night. In the initial map we placed the

stars at a medial position and observed that the DFM algorithm corrects this model by adding components

with positive intensity at the real position of the star. Simultaneously the algorithm adds δ functions with

negative amplitude at the place of the stellar disks in the initial model. Since the size of the beam may

change, it is difficult to compare the resulting difference maps directly. It is more promising to compare

residual difference maps obtained by subtracting a convolution of the initial model with the CB from the

DFM for each night. Figure 7.8 shows the residual difference maps for all five nights of our observation

run in November 2002. Since in some maps the residuals are only 10% we see several noise features. The

clockwise movement of the Aa component appears significantly in the lower right quadrant following the

color sequence black → red → blue → green → magenta. Less significant (due to the shorter arc) appears

the movement of the Ab components in the upper left quadrant.

—81—

Tabl

e7.

1:Pr

oper

ties

ofth

ege

nera

ted

map

sof

Cap

ella

.M

JD=

JD-2

4525

00

Dat

afr

omN

ight

Map

ping

I Ab/I

Aa

Stel

lar

Dia

m.

[mas

]Po

sitio

n[m

as]

Ref

.Po

s.(H

um94

)D

ate

MJD

Alg

orith

mC

omp.

DA

aD

Ab

dRA

dDE

CdR

AdD

EC

11/1

5/02

93.7

97C

HM

1.67

15.3

3−

39.8

617

.33

−39

.72

...94

.042

11/1

2/02

90.8

21C

HM

1.72

9.2±

4.8

6.5±

3.0

14.9

5−

40.2

014

.05

−41

.86

...11

/16/

02...

95.0

3411

/12/

0290

.821

DFM

1.70

10.3±

3.0

6.0±

3.8

15.0

5−

40.4

714

.05

−41

.86

...11

/16/

02...

95.0

34

Dat

afr

omN

ight

Dat

aM

appi

ngSt

eps

Rot

.B

eam

Size

Figu

reD

ate

MJD

Poin

tsA

lgor

ithm

[mas

]

11/1

5/02

93.7

9712

3/12

2/12

3C

HM

75no

6.8×

3.7

7.3

...94

.042

11/1

2/02

90.8

2133

6/28

7/32

8C

HM

80ye

s5.

4×

2.6

7.4

...11

/16/

02...

95.0

3411

/12/

0290

.821

336/

287/

328

DFM

200

yes

5.4×

2.6

7.5/

7.6

...11

/16/

02...

95.0

34

Tabl

e7.

2:Pr

oper

ties

ofth

ege

nera

ted

map

sof

λV

ir.

MJD

=JD

-245

2500

Dat

afr

omN

ight

Dat

aPo

ints

Map

ping

Step

sB

eam

Size

I Ab/I

Aa

Posi

tion

Figu

reD

ate

MJD

NA

lgor

ithm

[mas

]dR

A[m

as]

dDE

C[m

as]

03/2

2/03

...03

/24/

0321

9.85

2...

222.

937

89/1

73/1

57C

HM

309.

1×

2.4

2.6

16.8

41.

447.

7

7.2 Results

CHM - Capella (2002 November 15)

a) Contours (5% interval) b) Surface Plot

-40 -20 0 20 40

-40

-20

0

20

40

dRA [mas]

2002 November 15

0

0.5

1

c) Map d) uv-plane coverage

-5 0 5

-10

0

10

2002 November 15

Figure 7.3: Hybrid map of Capella using the data from November 15, 2002. For more details see table 7.1. The differentcolors in the uv-plane plot correspond to the different baselines: Black=AC; Red=BC; Blue=AB.

—83—


CHM - Capella (2002 November 12..16)


-40 -20 0 20 40

-40

-20

0

20

40

dRA [mas]


0

0.5

1


-10 0 10

-20

-10

0

10

20


Figure 7.4: Hybrid map of Capella using the complete data from 2002 November 12..16. For more details see table 7.1.The different colors in the uv-plane plot correspond to the different baselines: Black=AC; Red=BC; Blue=AB.

—84—

7.2 Results

DFM - Capella (2002 November 12..16)


-40 -20 0 20 40

-40

-20

0

20

40

dRA [mas]


0

0.5

1


-10 0 10

-20

-10

0

10

20


Figure 7.5: Difference map of Capella using the complete data from 2002 November 12..16. For more details seetable 7.1. The different colors in the uv-plane plot correspond to the different baselines: Black=AC; Red=BC; Blue=AB.

—85—


DFM - Capella (2002 November 12..16)

Vis. and Phases from Initial Model Vis. and Phases after DFM

500 600 700 800 900

500 600 700 800 900

File Number

500 600 700 800 900

500 600 700 800 900

File Number

Figure 7.6: The initial model (left) and the corrected model (right) for the Difference map of Capella shown in figure7.5.The Self-Calibration process adjusted the phases ϕBC (blue), ϕAC (green) and ϕBA (cyan). Measured visibilities are blackwhereas the visibilities calculated from the MM are red.

—86—

7.2 Results

CHM - λ Vir (2003 March 21..24)


-40 -20 0 20 40

-40

-20

0

20

40

dRA [mas]

2003 March 21..24

0

0.5

1


-10 0 10

-10

0

10

2003 March 21..24

Figure 7.7: Hybrid map of λ Vir using the complete data from our observation run in 2003 March 21..24. For moredetails see table 7.1. The different colors in the uv-plane plot correspond to the different baselines: Black=BC; Red=AC;Blue=BA.

—87—


-40 -20 0 20 40

-40

-20

0

20

40

dRA [mas]

Residual Difference Map 2002 November 12..16

Figure 7.8: Superposition of residual difference maps for five individual nights. Each map was obtained using exactlythe same initial model. To obtain the residuals between this initial model and the resulting difference map, we convolvedthe initial model with the CB and subtracted it from the final difference map. Following the used color sequence (Black:Nov12; Red: Nov13; Blue: Nov14; Green: Nov15; Magenta: Nov16) one can see the movement of the stars. Close tothe position of the stellar disks in the initial model the DFM algorithm removed intensity by adding δ functions withnegative amplitude.

—88—

Chapter 8

Remarks on the Performance of the

IOTA Facility

To summarize our experience from both observation runs and our efforts for data reduction, we want to make

the following remarks about the current performance of the IOTA observatory. In some cases we point out

potential problems and make constructive suggestions for future improvements of this remarkable facility.

Naturally we cannot incorporate most recent modifications which were carried out since our last observation

run in March 2003.

• During an engineering period, the “resonance effect” should be investigated and eliminated if possible.

We had found that the influence of this effect may be reduced with the development of algorithms

other than the power spectrum method. But presumably all results obtained would improve when the

resonance problem is eliminated.

• Arguably, the largest imbalance between the telescopes comes from an unequal positioning of the

fibers on the camera-pixels. It is part of the IOTA-observation procedure to use the ”Fiber Explorer”

software to maximize the flux on the PICNIC CCD-pixels. During the observation run in March

2003, we realized that a multiple repetition of this procedure during the night may improve the results

significantly. So this could be an affect of temperature variations during the night. Also refraction will

effect the alignment of the fibers on the camera. The refraction coefficient depends on the wavelength,

so with changing altitude the star tracker will continue to track the optical position while the position

—89—

Chapter 8 Remarks on the Performance of the IOTA Facility

of the infrared peak on the CCD moves. In low-altitude observations, another way to get an inequality

in the intensities from the different telescopes would be a reduction of the effective mirror area caused

by partial occultation of the siderostats by the domes.

• Currently, the matrixfiles which are intended to be used to correct for those imbalances are inadequate

for the requirements. Quite often the matrixfiles have to be rejected because the star tracker was not

able to track the star correctly. A useful modification would be to include a data reduction algorithm

into the data acquisition software. This algorithm should determine the flux ratio measured between

the telescopes and recommend the observer to repeat the acquisition of the matrixfiles automatically

if necessary. The computation time for such an algorithm is completely negligible.

• The fringe tracking algorithm based on the FFT could be replaced by a wavelet based algorithm. This

may improve the performance especially in cases with bad SNR. The same holds for the function

which allows to scan with the short delays for fringes. Since the computation of the CWT is more

expensive than the computation of a single FFT, it must be examined if the current hardware equipment

allows to run this CWT-algorithm on the IOTA site in real-time. The algorithm I implemented in the

data reduction software is based on the computation of FFTs (Tor98) with an overall complexity of

O(Ndata f ile · log(Ndata f ile)). There is a faster but also much more complex algorithm based on B-

Splines (Muñ00) which follows O(Ndata f ile). Another very simple way to reduce the computation

time is to narrow the range of computed scales close around the fringe.

• The filter algorithm presented in chapter 5.3.3 (see figure 5.12) could be implemented into the data

acquisition software to present the user with the noise filtered signal on screen during acquisition.

This would simplify the manual localization of the fringe and allow the user a better judgement of the

quality of the data.

• A data reduction program should be installed on one of the computers on site to allow the observer to

make a first appraisal of the quality of the data already during the observation run. This could help

judging the extent to which extend an object is resolved and how good the closure phase measurements

for this object are. In particular, the second point seems to be a limiting factor for the aperture synthesis

capabilities of the instrument. Typically estimations of fast changing closure phases show large errors.

So it would be better to adjust the number Ndata f ile if the closure phase is changing very rapidly or to

split datafiles later during the data reduction process to improve the closure phase estimations.

—90—

Chapter 9

Conclusions

The goal of my Masters project was to develop a general data reduction procedure for IOTA interferograms,

to confirm the stability of the instrument and to give IOTA the capability of imaging. Since the development

of a reliable data reduction procedure would have a huge influence on the other two tasks, I placed the highest

emphasis on a proper visibility and closure phase estimation and implemented two different algorithms

beyond the current standard routine. Both approaches provide remarkably similar results and each seem to

be superior for different tasks.

The Fringe Envelope Fitting seems to be superior for less resolved objects and may be used for the estimation

of stellar diameters as demonstrated for the resolved calibrators α Cas and α Lyn in chapter 6.2. For α Cas

the result agrees well with earlier published works, whereas the α Lyn fit is based on only one measurement

and seems to be affected by the worse conditions of this one measurement.

The CWT approach provides excellent results for low visibilities and we used those for model fits which

resulted in high precision positions of the Capella giants, confirming earlier published orbital elements. In

addition, we measured the diameters and the intensity ratio of the components for the first time in the H

band. Those results are also in excellent agreement with other diameter measurements and the intensity ratio

extrapolated from observations at shorter wavelengths. Based on the above we derived physical parameters

like the densities and effective temperatures of the individual components.

Finally, we presented maps of the binary stars Capella and λ Vir. By performing a coordinate transformation

we were able to compensate for the revolution of the Capella components during our observation run using

the known orbit. The resulting map has excellent resolution and a beam size of just a few mas. For mapping

—92—

we used the Difference Mapping strategy which was previously used by (You99) for imaging purposes at

optical and infrared wavelengths. Beside reaching this milestone for the IOTA facility we were also able,

as far as known, to construct the first infrared maps based on the completely model independent Hybrid

Mapping strategy.

The point of the Capella campaign was to image an object with known properties to demonstrate that the

IOTA is a capable imaging instrument. We have also gone beyond this basic objective by imaging the λ Vir

system for the first time. The disks of those stars are not resolved by IOTA but it will be straight forward

to include other observations by the IOTA collaboration to obtain an orbit. We measured the position and

intensity ratio of both components for the observed epoch and generated hybrid maps. Since we did not

measure any significant movement of the components over the four-day observation period, we can reject

the possibility that the 1.93017d period reported by (Abt61) and (Tok97) represents the orbital period of the

observed ≈ 20mas separated close binary system.

Within chapter 8 we made general remarks on the current performance of the IOTA observatory and con-

firmed that IOTA is able to measure visibilities with a precision of the order of 5% and better. To enhance

this performance we presented in section 5.4 an algorithm to correct for potential imbalances between the

telescopes. The proper behavior of this correction must still be demonstrated since the quality of our matrix-

files was not adequate for this purpose. The precision of the measured closure phases is of the same order

even if fast changing phases are problematic.

Going ahead from this point, IOTA is one of the few facilities with imaging capabilities which may be

used for further studies of close or interacting multiple stars or for the imaging of stellar surfaces and disk

features around young stellar objects (YSO).

—93—

A - Observation Logs

—94—

Appendix A

Table 9.1: Observation Log November 2002.

Date [UT] Time [UT] Hour Angle [h] Source Filter No. of data fileof first data acquisition (without matrix file)s

BASELINE: A=ne35, B=se15, C=ne011/11/02 06:03 1.32 α Cas H 0-1

08:00 -1.32 Capella H 6-708:34 -1.47 β Aur H 12-1310:57 1.63 Capella H 18-51, 56-6011:58 1.94 β Aur H 61-6813:08 3.82 Capella H 73-77

11/12/02 05:41 1.03 α Cas H 0-707:43 -1.54 Capella H 11-3109:11 -0.78 β Aur H 36-4310:11 0.94 Capella H 48-7511:15 1.28 β Aur H 80-8912:00 2.76 Capella H 94-11013:22 3.41 β Aur H 115-119

11/13/02 03:58 3.35 α Cyg H 0-1604:51 0.25 α Cas H 21-3306:42 -3.20 δ Aur H 38-4707:19 -1.87 Capella H + 25%-ND 52-7808:22 -1.53 β Aur H 83-9209:09 -0.04 Capella H + 25%-ND 97-11710:08 0.24 δ Aur H 122-13210:45 1.57 Capella H 137-16111:42 1.80 β Aur H 166-17512:20 2.44 δ Aur H 180-18913:14 -0.02 α Lyn H 198-207

11/14/02 02:23 1.83 α Cyg H 0-1204:08 3.58 α Cyg H 17-2604:44 0.20 α Cas H 31-4005:25 -1.60 κ Per H (45)50-5405:51 -3.27 Capella H + 25%-ND 59-8406:54 -2.94 β Aur H 89-9807:34 -2.27 δ Aur H 102-11608:24 -0.72 Capella H + 25%-ND 121-14209:17 -0.55 β Aur H 147-15809:52 0.03 δ Aur H 163-17310:25 1.30 Capella H + 25%-ND 178-21711:37 1.79 β Aur H 22212:06 2.98 Capella H + 25%-ND 236-25513:04 3.25 δ Aur H 260-269

—95—

Appendix A

Table 9.2: Continue: Observation Log November 2002.

Date [UT] Time [UT] Hour Angle [h] Source Filter No. of data fileof first data acquisition (without matrix files)

BASELINE: A=ne15, B=se15, C=ne011/15/02 06:45 2.30 α Cas H (0)10-15

07:08 3.36 Capella H + 25%-ND 20-4507:59 -1.07 | 1.65µ-ND 50-6108:21 -0.70 | H 66-8308:59 -0.79 δ Aur H 88-9709:28 0.42 Capella H + 25%-ND 102-15511:04 1.30 δ Aur H 160-16911:34 2.51 Capella H + 25%-ND 174-19212:08 3.09 | 1.65µ-ND 197-21512:50 3.80 | H + 25%-ND 220-22613:19 3.56 δ Aur H 231-238

11/16/02 08:14 -1.47 δ Aur H 0-1309:12 0.21 Capella H + ND 18-6311:07 1.42 β Aur H 68-8111:45 2.05 δ Aur H 86-10012:22 3.39 Capella H + ND 105-12213:05 3.39 δ Aur H 127-137

—96—

Appendix A

Table 9.3: Observation Log March 2003.


BASELINE: A=ne35, B=se7, C=ne2503/21/03 07:32 0.71 δ Crt H 0-20

08:18 -1.59 HD126035 H 25-2708:27 -1.36 λ Vir H 32-4009:05 -0.81 HD126035 H 45-4709:15 -0.57 λ Vir H 52-5510:21 0.47 HD126035 H 60-64

BASELINE: A=ne35, B=se7, C=ne1003/22/03 04:40 1.77 β CMi H 0-25

05:47 -0.99 δ Crt H 30-4706:23 0.77 λ Hya H 52-6406:48 0.04 δ Crt H 69-8008:11 -1.63 HD126035 H 85-8808:31 -1.23 λ Vir H 93-9808:41 -1.13 HD126035 H 103-10708:54 -0.86 λ Vir H 112-11309:25 -0.33 | H 118-14409:56 0.12 HD126035 H 149-15810:29 0.74 λ Vir H 163-18210:55 1.11 HD126035 H 187-19511:59 -0.91 HD158352 H 200-21512:32 -0.45 HD159170 H 220-22712:50 -0.06 HD158352 H 232-239

—97—

Appendix A

Table 9.4: Continue: Observation Log March 2003.


BASELINE: A=ne35, B=se15, C=ne1003/23/03 03:23 2.01 δ Aur H 0-8

03:59 2.44 HR2152 H 13-1704:19 2.94 δ Aur H 22-2704:40 3.12 HR2152 H 32-3705:35 1.06 75 Cnc H 42-5006:06 1.60 HR3621 H 55-6006:32 2.01 75 Cnc H 65-7006:55 2.42 HR3621 H 75-7907:45 1.06 δ Crt H 84-8908:51 -0.91 HD126035 H 95-9809:00 -0.68 λ Vir H 103-10709:09 -0.60 HD126035 H 112-11509:27 -0.23 λ Vir H 120-12809:41 -0.07 HD126035 H 133-13509:48 0.13 λ Vir H 140-16310:15 0.50 HD126035 H 168-17110:35 0.83 | H 176-17810:43 1.03 λ Vir H 183-19811:01 1.27 HD126035 H 203-21011:57 -0.84 HD157856 H 215-21912:08 -0.71 HD158352 H 224-23312:35 -0.33 HD159170 H 238-24412:56 0.10 HD158352 H 248-254

03/24/03 03:11 1.87 δ Aur H 0-504:15 3.66 Capella H 10-3206:17 -0.34 δ Crt H 37-4106:24 -0.22 | H 46-4806:41 0.05 | H 53-6807:24 -2.29 HD126035 H 73-7707:52 -1.76 λ Vir H 81-10408:20 -1.35 HD126035 H 109-11308:43 -0.90 λ Vir H 118-14309:16 -0.43 HD126035 H 148-15209:37 0.00 λ Vir H 161-18310:05 0.40 HD126035 H 188-19110:13 0.61 λ Vir H 196-21210:35 0.90 HD126035 H 217-22211:03 -1.52 41 Oph H 227-23111:09 -1.42 | H 235-23911:31 -1.25 HD158352 H 244-26212:17 -0.43 HD157856 H 267-27112:30 -0.27 HD158352 H 276-301

—98—

B - Data Reduction Software Manual

We provide a brief overview about the functionality of the developed data reduction software. All programs

are written for Linux gcc and allow the complete data reduction as presented in this thesis. The software

package consists of three applications which are normally used in the following order:

IDRS: Estimates the visibilities and closure phases from the IOTA data files and writes them into an outputfile.

ModelBin: Fits binary models to the IDRS output file.

Map: Produces maps from the IDRS output files and allows to analyze those maps.

The programs are developed for functionality and are not consequently optimized with respect to the

computation time. When used to process data for publication, we ask to give credit to this thesis. In the

following we present the different programs in more detail.

—99—

Appendix B

B.1 IDRS - Interferometry Data Reduction Software

IDRS estimates visibilities and closure phases using one of the three presented algorithms. To select

which algorithm is used and to enable different corrections, there is a declaration section within the program

file idrs.c that contains beside others the following parameters:

fitampmethod: Selects the algorithm for visibility estimation.

docalib: Enables the imbalance correction using the matrix files.

dobiascorr: Enables the BIAS correction for the fringe envelope fitting algorithm as formalized in equa-tion (5.6).

simfringe: Generates simulated fringes instead of using the data. Further parameters must be set withinthe code.

calibmethod: Allows to switch between averaging of the transfer function over the whole night and theinterpolation with equation (5.3).

doplot1..3: Enables up to three PGPLOT windows which show detailed plots during the data reductionprocess. These options may be set for debugging and confirming the correct performance of thealgorithms.

repdetails: Reports details of the running process on screen.

repfiles: Writes several report and debugging files on disk. Those files may be used to generate miscel-laneous plots with SM or MATLAB.

filterdata: Filters the obtained visibility and closure phase data using different error thresholds andscattering criteria.

The declaration section is grouped into subsections where all parameters are collected that affect the

performance of the different algorithms. The user may check out those sections for finetuning. After setting

those parameters and compiling the code, the user must generate a text file which lists the location of the

IOTA data file on the local hard disk. The following file nov02.ini may be used as example:

/home/skraus/iota/Data/0 /home/skraus/iota/Data/2002Nov15/ 0 0 0 00 /home/skraus/iota/Data/2002Nov16/ 1 2 50 200

Using this file, IDRS will assume that /home/skraus/iota/Data/ (called [DataRootDir]) contains fur-

ther initiation files and will process all data files in directory

/home/skraus/iota/Data/2002Nov15/ and scans 50 to 200 within the files iota1.data and iota2.data

in the directory /home/skraus/iota/Data/2002Nov16/.

—100—

Appendix B

The initiation files assumed in [DataRootDir] are text files named objid.ini and sre.ini. objid.ini

correlates the Object Designations in the IOTA header files with the IDRS intern pure numerical object ID,

for example

13Alp_Aur -> ID 134Bet_Aur -> ID 233Del_Aur -> ID 318Alp_Cas -> ID 440Alp_Lyn -> ID 527Kap_Per -> ID 650Alp_Cyg -> ID 7

The file sre.ini specifies which of those objects serves as calibrator and which may be used as target for

stellar radius estimation (SRE) as described in chapter 6.2. Objects which are targets but not used for SRE

are not listed! The file may look like

Calibrators: 2 3 6 7Targets: 4 5Diameters: 4.68 6.46

The program runs through a cycle which consists of the reduction of the matrix files, the visibility estima-

tion, the closure phase estimation and the SRE which is followed by the filtering of the data. The program

itself can be started using different command line parameters to skip some of those stations assuming that

the earlier cycles were already processed

./idrs comp -f nov02.ini: Performs the complete data reduction cycle as described above on thedata specified in the initiation file nov02.ini.

./idrs vis -f nov02.ini: Starts the cycle with the visibility estimation.

./idrs cp -f nov02.ini: Starts the cycle with the closure phase estimation.

./idrs sre 4 -f nov02.ini: Performs only the SRE on the object with ID 4 and finishes with thefiltering step.

The major output file of IDRS which is processed by the other applications is called vis.dat and includes

all measured visibility and closure phase data.

—101—

Appendix B

B.2 ModelBin - Fitting Binary Models

This application models visibilities and closure phases for binary sources and applies a least square fit

to the IDRS output file. The file general.h defines with the parameters MJDfrom, MJDto and target_id

which object should be fitted and what’s the temporal window for the fit (MJD=JD-2452500). Important

parameters for the fit are weightfunc1 and weightfunc2 which correspond to the parameters WV and

WΦ in chapter 6. max_guess and step_guess determine what the sizes of the initial guess grid and its

spacing are. With Iratio_from, Iratio_to and Iratio_step one may perform the fit for a whole range

of intensity ratios. This will produce the output file intratio.dat. Starting the program with the command

line parameter ./modelbin intratio a parabola is fitted to the obtained χ2(I2/I1) ratios and the application

will provide the best fit intensity ratio with an error estimation.

To perform a diameter fit of the disks as well, one has to fix the positions of the stars with fitdiameter_dRA,

fitdiameter_dDEC and enable this fit with the option fitdiameter.

The program can also be used to rotate the (u,v) plane as described in chapter 2.8.1. The corresponding

orbital elements have to be set in the file model.c and the rotation itself is performed with ./modelbin

rotate. The rotated (u′,v′) coordinates are written within the file vis.dat such that map will use the rotated

data for mapping. To restore the unrotated data one can just copy the file visnorot.dat to vis.dat.

B.3 Map - Mapping Software

Our Mapping Software uses the DFT and is therefore quite inefficient for large data files. The map is

produced for the object and with data within the temporal window as defined in general.h. Important

parameters within the declaration section are clean_gain and max_cleansteps which specify the gain

and the number of clean steps performed in each CLEAN cycle (reasonable values are 0.02 as gain and

400 cleansteps for CHM or 20 cleansteps for DM). Beside that the user may choose between two rect-

angular or two circular clean windows by setting clean_window_type. hybridsteps determines how

many CHM or DM steps are performed before the iteration ends. For CHM one may use use_model_vis,

use_model_phase and use_model_phase_guess either to use only modeled visibility and closure phase

data or to use the model phases from modelbin as initial guess for the hybrid map on the real data. For

DM the parameters model_I1, model_I2, model_dRA, model_dDEC, model_R1 and model_R2 must be set.

—102—

Appendix B

The dimensions of the maps in pixels are set with size_x and size_y where the conversion factor from

pixel to mas is given by spacing. To start mapping the application is started with ./map hybrid or ./map

diff. The iteration can be interrupted by the user to end mapping or to relocate the clean windows, which

is done directly in the PGPLOT window. The final maps are named map.dat and mapbeam.dat where the

latter contains the beam in the corner of the map. The output format is float binary with two integers in the

beginning of the file giving the dimensions along both axes.

After a map is generated successfully, it may be analyzed running ./map analyse. This part of the pro-

gram allows to select two regions. After this selection the fluxes within those regions and the center of

light-positions are calculated. With ./map center one may center one star or the center of light in the

middle of the map. The output file with centered star is called cenmap.dat. Analogous performs the com-

mand ./map diameter a convolution of a uniform disk model with the dirty beam (from file db.dat) and

measures the response to a star in the map map.dat. This routine generates a output file with the χ2 values

for different diameters which may be used by ./modelbin intratio to obtain the best-fit diameter and its

error by performing a Brute Force Fit.

—103—

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